TRANSACTIONS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 351, Number 2, February 1999, Pages 429–475
S 0002-9947(99)02335-1
QUADRATIC HOMOLOGY
HANS-JOACHIM BAUES
Abstract. We describe axioms for a ‘quadratic homology theory’ which generalize the classical axioms of homology. As examples we consider quadratic
homology theories induced by 2-excisive homotopy functors in the sense of
Goodwillie and the homology of a space with coefficients in a square group
which generalizes the homology of a space with coefficients in an abelian group.
0. Introduction
The development of algebraic topology was profoundly affected by the notion of
homology. Originally homology had coefficients in abelian groups and EilenbergSteenrod described the axioms of such an ordinary homology theory. Somewhat
later important examples of generalized homology theories were found which led to
the notion of homology with coefficients in a spectrum. The spectrum-homology
can also be described as a homology theory satisfying all Eilenberg-Steenrod axioms
except the dimension axiom concerning the value of the homology on spheres. In
fact this value characterizes a homology theory in the sense that a natural transformation of homology theories which is an isomorphism on spheres is also an
isomorphism on all finite CW-complexes.
In his recent work on the calculus of homotopy functors Goodwillie observed
that a spectrum is equivalent to a linear homotopy functor and spectrum-homology
of a space X can be equivalently described by the homotopy groups.
(1)
Hn (X, L) = πn L(X)
where L is a linear homotopy functor. We therefore call spectrum-homology also
a linear homology theory. The non-linear homology theories are then obtained by
the homotopy groups
(2)
Hn (X, D) = πn D(X)
where D is any homotopy functor. Hence we again generalize homology by choosing
now homotopy functors as coefficients. Clearly this is a very far reaching generalization since in particular homotopy groups of a space are the homology groups
with coefficients in the identity functor. It is an old problem to find axioms which
characterize the theory of homotopy groups in a similar way as ordinary homology
theory is characterized by the Eilenberg-Steenrod axioms. The identity functor is
Received by the editors October 22, 1996.
1991 Mathematics Subject Classification. Primary 55N35, 55Q70, 55S20.
Key words and phrases. Quadratic functors, Goodwillie calculus, Steenrod squares, EHPsequence.
c
1999
American Mathematical Society
429
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430
HANS-JOACHIM BAUES
not linear but is still an analytic functor in the sense that there is a Taylor tower,
n ≥ 1,
(3)
X → Pn (X) → Pn−1 (X) → . . . → P2 (X) → P1 (X)
approximating X. Here P1 is linear, P2 is quadratic and more generally Pn is a
reduced and n-excisive homotopy functor; compare [19, 20]. A reduced n-excisive
homotopy functor from spaces to spaces is the topological analogue of a polynomial
functor of degree n in algebra. The linear functor P1 yields the homology theory
of stable homotopy groups πnS (X) = Hn (X, P1 ).
In this paper we study as a first step outside the linear world the quadratic
homology theories Hn (X, Q) obtained by a quadratic homotopy functor Q. We
introduce axioms of a quadratic homology theory such that Hn (X, Q) satisfies these
axioms. As in the classical case our axioms characterize a quadratic homology
theory in the sense that a natural transformation between theories which is an
isomorphism on spheres is also an isomorphism on all finite CW-complexes. We
deduce from the axioms various facts like the general EHP-sequence in §3. For
example quadratic homotopy groups
(4)
πnQ (X) = Hn (X, P2 )
defined by P2 in the Taylor tower (3) satisfy all the axioms. This is the quadratic
analogue of stable homotopy groups.
The main purpose of this paper is a first discussion of basic axioms of a “quadratic homology theory” and the computation of examples obtained by “square
homology”. A problem like the quadratic analogue of Brown’s representability
theorem remains open; see the remark following (4.7).
Let Gr be the category of groups. Then any group functor F : Gr → Gr as
considered in (5.3) below induces a homotopy functor F] which carries connected
spaces to connected spaces. If F is linear, that is, if the cross effect of F is trivial (see
2.9) then F (G) = Gab ⊗ A is given by an abelian group A and ordinary homology
with coefficients in A can be described by
(5)
H̃n (X, A) = Hn (X, F] ).
If the group functor F is quadratic, that is, if the cross effect of F is bilinear (see
(2.9)) then we know by [10] that F (G) = G ⊗ M is given by a square group M
which is the quadratic analogue of an abelian group. We prove that then F] is a
quadratic homotopy functor. Hence the square homology
Hn (X, M ) = Hn (X, F] )
with coefficients in a square group M is a quadratic homology theory which generalizes ordinary homology. Using the Taylor tower of F] we obtain the linearization
(7)
F] = P2 F] → P1 F] = F]lin
which gives us a linear homology theory
(8)
HnS (X, M ) = Hn (X, F]lin )
termed stable square-homology. Here F]lin corresponds to a spectrum EM so that
we get a functor from the category of square groups to the homotopy category of
spectra which carries M to EM . We show that EM is always the cofiber of a map
SqM which carries an Eilenberg-Mac Lane spectrum to a product of Eilenberg-Mac
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QUADRATIC HOMOLOGY
431
Lane spectra. We call SqM the squaring operation associated to M . For example
all Steenrod squares Sq 2 , Sq 3 , . . . can be derived from SqM .
Let Γn G and Γ̄n G be the subgroups of a group G given by the lower central
series and the mod-2 restricted lower central series respectively. Then we obtain
the quadratic group functors
nil2 , nil2 : Gr → Gr
which carry G to the quotients G/Γ3 G and G/Γ̄3 G respectively. The corresponding
square groups are Znil and Z4,2
nil with
nil2 (G) = G ⊗ Znil ,
nil2 (G) = G ⊗ Z4,2
nil .
We compute the squaring operation SqM for M = Znil and M = Z4,2
nil explicitly
in terms of Steenrod squares. This determines the spectrum associated to the
linearization of the homotopy functor (nil2 )] , resp. (nil2 )] ; see (8.16), (8.17).
The author happily acknowledges helpful conversations with Teimuraz Pirashvili.
He also thanks G. Arone and T. Goodwillie for comments concerning “Calculus”.
1. CW-spaces, CW-pairs and spectra
We use the following conventions: Top is the category of topological spaces and
Top∗ is the category of topological spaces with base point. We obtain cofibrations
in Top by the universal homotopy extension property defined via the cylinder
I × X where I = [0, 1] is the unit interval. A homotopy in Top∗ is a pointed map
I(X) → Y where I(X) = I × X/I × {∗} is the reduced cylinder. By the inclusion
(i0 , i1 ) : X ∨ X ⊂ I(X) we obtain the cone CX = I(X)/i1 X and the suspension
ΣX = CX/i0 X. Here we use the quotient space X/Y which is defined for any pair
of spaces (X, Y ) by the adjunction X/Y = X ∪Y {∗}. More generally an adjunction
space X ∪Y Z is defined by a push out diagram
X −−−−→ X ∪Y Z
ḡ
x
x




g
Y −−−−→
Z
If (X, Y ) is a pair of spaces we call ḡ : (X, Y ) → (X ∪Y Z, Z) an adjunction map.
A space X is a CW-space if there is a CW-complex Y together with a homotopy
equivalence Y ' X in Top. The space X is a finite CW-space if Y can be chosen
to be a CW-complex with finitely many cells. Moreover we write dim(X) ≤ n if
dim(Y ) ≤ n. A CW-pair (X, Y ) is a pair in Top of CW-spaces X and Y for which
the inclusion Y ⊂ X is a cofibration. The pair is a finite CW-pair if X and Y are
finite CW-spaces. A CW-space X is well pointed if the inclusion {∗} → X is a
cofibration.
Let space ⊂ Top∗ be the full subcategory of well pointed CW-spaces and
pointed maps. Moreover let pair be the category of well pointed CW-pairs and
pointed pair maps. We have functors
space → pair → space
which carry the space X to the pair (X, ∗) and carry the pair (X, Y ) to the quotient
space X/Y . We also use the full subcategory space r of (r − 1)-connected objects
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432
HANS-JOACHIM BAUES
in space and the full subcategory pair r of objects (X, Y ) in pair for which X and
Y are (r − 1)-connected. We point out that the suspension yields a functor
Σ : space r → space r+1
raising the degree of connectedness.
A spectrum E is a sequence of maps εn : ΣEn → En+1 in space, n ∈ Z. A
map f : E → E 0 between spectra is a sequence of maps fn : En → En0 with
fn+1 εn = ε0n (Σfn ). Let spectra be the category of such spectra and maps.
A map f : X → Y in Top is a weak equivalence if f induces a bijection of
homotopy groups f∗ : πn X ≈ πn Y, n ≥ 0, for every basepoint in the domain. A
map in Top∗ is a weak equivalence if it is one in Top. Clearly a weak equivalence
in space or pair is also a homotopy equivalence. A map f : E → E 0 in spectra
is a weak equivalence if it induces an isomorphism f∗ : πk E ≈ πk E 0 . Here πk E =
colim {πn+k En } is an abelian group for all k ∈ Z.
2. Axioms of homology
To fix notation we recall the following basic properties of a homology theory.
(2.1) Definition. Let r ≥ 0. A suspension theory (H, s) is a sequence of covariant
functors, n ∈ Z,
Hn : space r → Ab
together with a sequence of natural transformations, n ∈ Z,
sn : Hn → Hn+1 ◦ Σ
such that Hn (∗) = 0 and Hn (f0 ) = Hn (f1 ) for homotopic maps f0 ' f1 . We also
consider suspension theories which satisfy some of the following axioms.
Exactness. For (X, Y ) ∈ pair r the sequence
i
q∗
∗
Hn (Y ) −→
Hn (X) −→ Hn (X/Y )
is exact where i : Y → X is the inclusion and q : X → X/Y is the quotient map.
Suspension. For X ∈ space r the natural map
sn (X) : Hn (X) → Hn+1 (ΣX)
given by the transformation sn is an isomorphism for all n ∈ Z.
Colimit axiom. For each sequence X0 X1 . . . of cofibrations in space r the
induced map
colim {Hn (Xi )} → Hn (colim {Xi })
is an isomorphism.
A suspension theory (H, s) is termed a homology theory on space r if the exactness and suspension axioms are satisfied; compare [29].
Given a supension theory (H, s) we obtain the stable theory (H S , s) associated
to (H, s). Here HnS (X) is the colimit of
(2.2)
s
s
Hn (X) −→ Hn+1 (ΣX) → . . . → Hn+k (Σk X) −→ . . . .
Clearly one obtains the canonical map
S
s : HnS (X) → Hn+1
(ΣX)
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QUADRATIC HOMOLOGY
433
which is an isomorphism. Hence (H S , s) is a suspension theory satisfying the suspension axiom. If (H, s) satisfies the exactness or colimit axiom then so does
(H S , s). Clearly for a homology theory we have (H, s) = (H S , s).
(2.3) Definition. Let r ≥ 0. A boundary theory (H, ∂) is a sequence of covariant
functors, n ∈ Z,
Hn : pair r → Ab
together with a sequence of natural transformations ∂ : Hn+1 (X, Y ) → Hn (Y ) with
(X, Y ) ∈ pair r and Hn (Y ) = Hn (Y, ∗), such that Hn (∗) = 0 and Hn (f0 ) =
Hn (f1 ) for homotopic maps f0 ' f1 in pair r . Moreover ∂-exactness is satisfied,
that is
∂
i
j∗
∂
∗
Hn (X) −→ Hn (X, Y ) −→ . . .
. . . → Hn+1 (X, Y ) −→ Hn (Y ) −→
is exact. Here i∗ and j∗ are induced by i : Y ⊂ X and j : (X, ∗) ⊂ (X, Y ). We also
consider boundary theories with the following additional property.
Excision. For (X, Y ) ∈ pair r and g : Y → Z ∈ space r the adjunction map ḡ
induces an isomorphism
ḡ∗ : Hn (X, Y ) ∼
= Hn (X ∪Y Z, Z)
for all n ∈ Z.
A boundary theory (H, ∂) is termed a homology theory on pair r if excision is
satisfied. Compare [29].
Each boundary theory (H, ∂) yields a suspension theory (H, s) as follows. Let
CX be the cone on X. Then ∂ : Hn+1 (CX, X) ∼
= Hn (X) is an isomorphism. Hence
we obtain the suspension map
s = q∗ ∂ −1 : Hn (X) ∼
(2.4)
= Hn+1 (CX, X) → Hn+1 (ΣX)
where q : (CX, X) → (ΣX, ∗) is the quotient map. If (H, ∂) is a homology theory then q∗ ∂ −1 is an isomorphism since excision implies that q∗ : Hn (X, Y ) →
Hn (X/Y ) is an isomorphism. This leads to the following well known lemma. For
this we observe that suspension theories, resp. boundary theories form categories.
Morphisms are the natural transformations compatible with s, resp. ∂.
(2.5) Lemma. The category of homology theories (H, ∂) on pair r and the category
of homology theories (H, s) on space r are equivalent. The equivalence carries
(H, ∂) to (H, q ∗ ∂ −1 ).
These categories actually do not depend on r since the category of homology
theories (H 0 , s0 ) on space r is equivalent to the category of homology theories (H, s)
0
(Σr X).
on space. The equivalence carries (H 0 , s0 ) to (H, s) with Hn (X) = Hn+r
This shows that a homology theory on space is determined by its restriction to the
category space r for arbitrary large r ≥ 0. Such a statement will not be true for
quadratic homology theories below.
(2.6) Example. (A) Let A be an abelian group and let Hn (X, Y ; A) be the singular
homology of the pair (X, Y ) with coefficients in A. This is the classical homology
theory on pair.
(B) Let πn (X, Y ) be the relative homotopy group of the pair (X, Y ). Then (πn , ∂)
is a boundary theory on pair2 with πn (X, Y ) = 0 for n ≤ 1. Here we use (X, Y ) ∈
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434
HANS-JOACHIM BAUES
pair2 in order to obtain abelian groups πn (X, Y ), n ∈ Z. The associated stable
theory (πnS , s) is the homology theory on space 2 termed stable homotopy.
The spherical groups of a homology theory H on pair r are the groups
Hn (S r ), n ∈ Z, where S r is the r-sphere. The suspension isomorphism Hn (S r ) ∼
=
Hn+k (S r+k ) shows that the spherical groups determine the value of H on all
spheres in space r . Moreover the following uniqueness-lemma is well known: Let
φ : H → H 0 be a natural transformation of homology theories on pair r such that
φ is an isomorphism on spherical groups; that is, φ : Hn (S r ) ∼
= Hn0 (S r ) for n ∈ Z.
Then φ is an isomorphism
φ : Hn (X, Y ) ∼
(2.7)
= H 0 (X, Y )
n
for all finite CW-pairs (X, Y ) ∈ pair r and n ∈ Z. Compare for example (2.12) in
[28]. A similar uniqueness lemma is also true for the quadratic homology theories
below. Moreover we have the following representability theorem of E.H. Brown;
compare for example I.3.8 in [28]. Let H be a homology theory on space r . Then
there exists a spectrum E and a natural isomorphism
Hn (X) ∼
(2.8)
n ∈ Z,
= πn (E ∧ X) = Hn (X, E),
for all finite CW-spaces X in space r .
Next we describe the “partial suspension” and the “cross effect suspension” of a
boundary theory. To this end we have to introduce the following notation on cross
effects.
(2.9) Notation. Let Gr be the category of groups and let C be a category with zero
object ∗ and assume that sums (coproducts) X ∨Y exist in C. For objects X, Y ∈ C
one has the unique zero morphism 0 : X → ∗ → Y . Maps f : X → Z, g : Y → Z
determine (f, g) : X ∨ Y → Z. We have the retraction r1 : X ∨ Y → X and
r2 : X ∨ Y → Y with r1 = (1, 0) and r2 = (0, 1). Given a functor
F : C → Gr
satisfying F (∗) = ∗ we define the cross effect
(1)
F (X|Y ) = kernel {(F r1 , F r2 ) : F (X ∨ Y ) → F (X) × F (Y )}.
This yields the functor F ( | ) : C × C → Gr with induced maps denoted by (f |g)∗ .
The inclusion i12 : F (X|Y ) ⊂ F (X ∨ Y ) is natural in X and Y . Moreover we have
the natural interchange isomorphism
∼ F (Y |X)
(2)
T : F (X|Y ) =
induced by T : X ∨ Y = Y ∨ X with T = (i2 , i1 ). Let F (X ∨ Y )2 be the kernel of
F r2 : F (X ∨ Y ) → F (Y ). Then we have the split short exact sequence of groups
(3)
i
Fr
12
1
0 → F (X|Y ) −→
F (X ∨ Y )2 −→
F (X) → 0
If F is a functor C → Ab we thus have natural isomorphisms
(F i1 , i12 ) : F (X) ⊕ F (X|Y ) = F (X ∨ Y )2 ,
(4)
(F i1 , F i2 , i12 ) : F (X) ⊕ F (Y ) ⊕ F (X|Y ) = F (X ∨ Y )
which we use as identifications. The functor F : C → Gr is linear if F (∗) = 0 and
F (X|Y ) = 0 for all X, Y ∈ C; that is
(F r1 , F r2 ) : F (X ∨ Y ) ∼
(5)
= F (X) × F (Y )
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QUADRATIC HOMOLOGY
435
is an isomorphism. The functor F : C → Gr is quadratic if F (∗) = 0 and the
cross effect F (X|Y ) as a bifunctor is linear in each variable X and Y . If F is linear
then the group F (X) is abelian and if F is quadratic then the group F (X) has
nilpotency degree 2. Moreover for a quadratic functor the subgroup F (X|Y ) is
central in F (X ∨ Y ). Compare [10].
Now let (H, ∂) be a boundary theory as in (2.3). Then one gets the following
exact sequences from which we derive the partial suspension E. Let CY be the
cone on Y so that Hn (CY ∨ X) = Hn (X). This yields the short exact sequence
r
∂
2
0 → Hn+1 (CY ∨ X, Y ∨ X) −→ Hn (Y ∨ X) −→
Hn (X) → 0
and hence the isomorphism
∂0 : Hn+1 (CY ∨ X, Y ∨ X) ∼
= Hn (Y ∨ X)2 .
Moreover one gets the short exact sequence
j∗
0 → Hn+1 (Y ) → Hn+1 (ΣX ∨ Y ) −→ Hn+1 (ΣX ∨ Y, Y ) → 0
which shows that j∗ induces the isomorphism
j0 : Hn+1 (ΣX ∨ Y )2 ∼
= Hn+1 (ΣX ∨ Y, Y )
Dividing out X yields the quotient map q ∨ 1 : (CX ∨ Y, X ∨ Y ) → (ΣX ∨ Y, Y ).
Now the partial suspension E is the composition
(2.10)
E = j0−1 (g ∨ 1)∗ ∂0−1 : Hn (X ∨ Y )2 → Hn+1 (ΣX ∨ Y )2 .
Compare [2, 3]. If Y = ∗ this is the suspension in (2.4). Moreover the partial
suspension restricts to cross effects yielding the cross effect suspension
s : Hn (X|Y ) → Hn+1 (ΣX|Y )
(2.11)
such that via (2.9) (4) the partial suspension is the composite
Hn (X ∨ Y )2
E
−−−−→
k
Hn+1 (ΣX ∨ Y )2
k
s⊕s
Hn (X) ⊕ Hn (X|Y ) −−−−→ Hn+1 (ΣX) ⊕ Hn+1 (ΣX|Y )
Hence for each Y the pair (Hn (−|Y ), s) is a suspension theory. Using the interchange map T in (2.9) (2) also (Hn (X|−), s̄) is a suspension theory where s̄ is the
composite
(2.12)
s̄ = T sT : Hn (X|Y ) ∼
= Hn (Y |X) → Hn+1 (ΣY |X) ∼
= Hn+1 (X|ΣY ).
The sum or coproduct in the category space r is obtained by the one point union
X ∨ Y of spaces.
(2.13) Lemma. Let H be a homology theory on pair r . Then Hn : space r → Ab
is linear, that is Hn (X|Y ) = 0 and Hn (X ∨ Y ) = Hn (X) ⊕ Hn (Y ).
Proof. For the CW-pair (X ∨ Y, Y ) ∈ pair r we have by exactness the split short
exact sequence
0 → Hn (Y ) → Hn (X ∨ Y ) → Hn (X ∨ Y, Y ) → 0
where Hn (i2 ) is injective since we have the retraction Hn (r2 ). Moreover excision
shows that Hn (X) → Hn (X ∨ Y, Y ) is an isomorphism.
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436
HANS-JOACHIM BAUES
(2.14) Lemma. Let (H, ∂) be a boundary theory on pair r . Then the suspension
s in (2.4) makes the following diagram commute
((r1 )∗ ,(r2 )∗ )
Hn (X ∨ Y )

s
y
−−−−−−−−→
Hn (X) × Hn (Y )

s×s
y
(i1 )∗ +(i2 )∗
Hn+1 (ΣX ∨ ΣY ) ←−−−−−−− Hn+1 (ΣX) × Hn+1 (ΣY )
This implies that the composition
i
s
12
Hn (X ∨ Y ) −→ Hn+1 (ΣX ∨ ΣY )
Hn (X|Y ) −→
is always trivial, si12 = 0.
Proof of (2.14). Let π : CX → ΣX be the quotient map. Then
π ∨1
1∨π
X
ΣX ∨ CY −→Y ΣX ∨ ΣY = Σ(X ∨ Y )
C(X ∨ Y ) = C(X) ∨ C(Y ) −→
is the quotient map πX∨Y . Hence we get the following diagram in which the row
is split short exact.
∂
Hn+1 (C(X ∨ Y ), X ∨ Y ) ∼
= Hn (X ∨ Y )
(πX ∨1)∗
0 → Hn+1 (ΣX)
/
Hn+1 (ΣX ∨ CY, Y )
∂
/
Hn (Y ) → 0
(1∨πY )∗
Hn+1 (ΣX ∨ ΣY )
Here we have the isomorphism (i1 )∗ + (i2 )∗
∼
=
Hn+1 (ΣX) ⊕ Hn+1 (CY, Y ) −→ Hn+1 (ΣX ∨ CY, Y )
by the exactness of the row. This yields the result.
3. Quadratic homology
In this section we introduce new axioms of a quadratic homology theory. These
axioms are satisfied by various examples which we describe in the following sections.
We deal with some consequences of the axioms, in particular we obtain the long
EHP-sequence of quadratic homology and we prove a uniqueness lemma.
(3.1) Notation. Let F : C → Gr be a functor where C is a category with zero
object ∗ and sums and let F (∗) = 0. We obtain the natural homomorphism
(1,1)∗
i12
P : F (X|X) ⊂ F (X ∨ X) −→ F (X)
for X ∈ C. Moreover if X is a cogroup in C with structure maps µ : X → X ∨ X
and ν : X → X satisfying the usual identities we define the function H, with
H
i12
F (X) −→ F (X|X) ⊂ F (X ∨ X),
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QUADRATIC HOMOLOGY
437
by i12 H(a) = F (µ)(a) − F (i2 )(a) − F (i1 )(a) where i1 , i2 are the inclusions of X
in X ∨ X. Clearly H is natural with respect to maps between cogroups. We write
F {X} for the pair of functions
H
P
F {X} = (F (X) −→ F (X|X) −→ F (X)).
It is shown in 3.6 [10] that F {X} is a square group (see (6.3) below) if F is a
quadratic functor. Clearly H is a homomorphism for a functor F : C → Ab.
If C is an additive category then a quadratic functor F : C → Ab yields for
X ∈ C a pair of homomorphisms F {X} = (H, P ) as above with HP H = 2H and
P HP = 2P ; that is F {X} is a quadratic Z-module; compare [5].
(3.2) Definition. Let r ≥ 0 and consider a boundary theory (Q, ∂) on pair r with
the following properties (i) and (ii).
(i) The stable theory QS associated to Q via the suspension (2.4) is a homology
theory.
(ii) For each Y ∈ space r the suspension theory of cross effects (Q(−|Y ), s) in
(2.11) is a homology theory.
A quadratic homology theory (Q, ∂, δ) is a boundary theory (Q, ∂) which satisfies
(i) and (ii) together with a sequence of homomorphisms, n ∈ Z,
δ : Qn+1 (CA ∪A X, X) → Qn−1 (A|A)
which are defined for all pairs (X, A) in pair r . The homomorphisms δ might look
obscure to the reader; yet the cubical diagram in the proof of (4.12) below (defined
for any quadratic homotopy functor) yields canonically such homomorphisms δ;
see (4.12) (5). The homomorphisms δ are natural in (X, A), that is: A pair map
f : (X, A) → (Y, B) ∈ pair r induces a commutative diagram
δ
Qn+1 (CA ∪A X, X) −−−−→ Qn−1 (A|A)



(g|g)
Qn+1 (Cg∪f )y
∗
y
δ
Qn+1 (CB ∪B Y, Y ) −−−−→ Qn−1 (B|B)
where g : A → B is the restriction of f . Moreover the following property (iii) is
satisfied.
(iii) For each pair (X, A), the sequence
δ
j]
. . . −→ Qn (A|A) −→ Qn+1 (CA ∨ X, A ∨ X)
i]
δ
−→ Qn+1 (CA ∪A X, X) −→ Qn−1 (A|A) → . . .
is exact. Here the inclusion i : A → X yields the map (i, 1) : A ∨ X → X
which defines the adjunction map (i, 1) : (CA ∨ X, A ∨ X) → (CA ∪A X, X)
which induces i] = (i, 1)∗ in the sequence. Moreover j] in the sequence is the
composition
j] = ∂0−1 (P, −(1|i)∗ ) : Qn (A|A) → Qn (A) ⊕ Qn (A|X)
∼ Qn+1 (CA ∨ X, A ∨ X).
= Qn (A ∨ X)2 =
Here P is the natural map in (3.1) and −(1|i)∗ is the negative of the induced
homomorphism (1|i)∗ = Qn (1|i). We call the long exact sequence (j] , i] , δ)
above the quadratic excision sequence.
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438
HANS-JOACHIM BAUES
It follows from (ii) above and (2.13) that all of the functors
Qn : space r → Ab
are quadratic. Moreover one can check that a quadratic homology theory (Q, ∂, δ)
for which these functors Qn are linear is the same as a homology theory on pair r .
This follows from (i) and (3.5) below.
Remark. Clearly axioms as described in (3.2) are obtained by choosing basic properties of the examples of quadratic homology theories which we consider below. We
have chosen the axioms in such a way that they are very close to the familiar concepts of a “boundary theory” (2.3), “cross effect” (3.1) and “EHP-sequence” (3.5)
and such that the uniqueness lemma (3.8) for these axioms is satisfied. A different
choice of axioms could be obtained by transforming Goodwillie’s definition of 2excisive homotopy functors; this however would involve functors defined on triads
(X; A, B) of spaces. In view of the theorem in Appendix A the use of such triad
groups is not necessary. It remains open to find a set of axioms for a quadratic
homology theory which implies the quadratic analogue of Brown’s representability
theorem; see the remark following (4.7). In any case the basic properties described
in (3.2) should be derivable from any good set of axioms describing “quadratic
homology theories”.
We now derive from the axioms of a quadratic homology theory some consequences. First we observe that the following diagram commutes.
Qn (A|A)
(3.3)
j#
/
Qn+1 (CA ∨ X, A ∨ X)
EE
EE
EE
∼
= ∂0
EE
EE
EE
Qn (A ∨ X)2
d2 EEE
EE
EE
EE
i#
/
Qn+1 (CA ∪A X, X)
∂
(i,1)∗
Qn (X)
/
"
Qn (A) ⊕ Qn (A|X)
/
d1
Qn (X)
where d2 = (P, −(1|i)∗ ) and d1 = (i∗ , P (i|1)∗ ). Here d1 d2 = 0 since P is natural;
in fact,
d1 d2 = i∗ P − P (i|1)∗ (1|i)∗ = i∗ P − P (i|i)∗ = 0.
Clearly d1 d2 = 0 also follows from i] j] = 0 and the commutativity of the diagram.
Using (3.3) we can equivalently describe the excision exact sequence in (3.2) (iii)
by the exact sequence in the row of the commutative diagram:
(3.4)
...
δ
/
Qn (A|A)
d2
/
Qn (A) ⊕ Qn (A|X)
d1
/
Qn+1 (CA ∪A X, X)
TTT
TTT
TTT
TTT
TTT
d1
T
δ
/
...
∂
)
Qn (X)
Here we set d¯1 = i] ∂0−1 by (3.4). This is similar to the diagram for metastable
homotopy groups in 2.4 of [4]. As a consequence of (3.4) we obtain a long exact
sequence which resembles the classical EHP-sequence of James [23].
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QUADRATIC HOMOLOGY
439
(3.5) Proposition. Let Q be a quadratic homology theory on pair r , r ≥ 0. Then
one has the following long exact sequence which is natural in A ∈ space r .
P
H0
s
P
. . . → Qn (A|A) −→ Qn (A) −→ Qn+1 (ΣA) −→ Qn−1 (A|A) −→ Qn−1 (A) → . . . .
This exact sequence is deduced purely algebraically from the axioms (3.2). Given
a quadratic homotopy functor D we always obtain the EHP sequence since (4.7)
below holds. The EHP sequence for the quadratic homotopy functor D can also be
deduced from Goodwillie’s classification of quadratic homotopy functors. In fact
D(X) is given as the fiber of a natural map Ω∞ (D1 ∧ X) → Ω∞ (D2 ∧ X ∧ X/Σ2 )
where D1 and D2 are spectra; see (8.2). As pointed out by the referee applying the
natural map D(X) → ΩD(ΣX) to this fibration yields as well the EHP-sequence
for D.
Proof. We consider (3.4) in case X = CA is the cone on A. Then the homotopy
axiom shows Qn (A|CA) = 0 and Qn (CA) = 0. Hence we obtain the isomorphism
i∗ : Qn+1 (CA ∪A CA) ∼
= Qn+1 (CA ∪A CA, CA)
where ΣA = CA ∪A CA. Thus (3.4) yields the commutative diagram
P
s
Qn (A|A) −−−−→ Qn (A) −−−−→
k
k
H0
−−−−→ Qn−1 (A|A)
Qn+1 (ΣA)
k
k
d¯1
d
δ
Qn (A|A) −−−2−→ Qn (A) −−−−→ Qn+1 (CA ∪A CA, CA) −−−−→ Qn−1 (A|A)
which defines the exact sequence in (3.5).
Given a natural exact sequence as in (3.5) one obtains the associated sequence
of cross effects which is also exact. Since si12 = 0 by (2.14) this yields the following
commutative diagram with exact rows.
σ0
(1,T )
H0
P
0 −−−−→ Qn+2 (ΣA|ΣA0 ) −−−−→ Qn (A|A0 )⊕Qn (A0 |A) −−−−→ Qn (A|A0 ) −−−−→ 0






y
y
y
Qn+2 (ΣA ∨ ΣA0 ) −−−−→ Qn (A ∨ A0 |A ∨ A0 ) −−−−→ Qn (A ∨ A0 )
Here the vertical arrows are the inclusions i12 . This shows that σ 0 induces the
isomorphism
(3.6)
σ : Qn+2 (ΣA|ΣA0 ) ∼
= Qn (A|A0 )
where σ = (pr1 )σ 0 is the composition of σ 0 and the projection pr1 . Clearly σ is
again natural in A and A0 .
Remark. Using the suspension isomorphism s̄ s in (3.2) (ii) and (2.11), (2.12) we
obtain a further natural isomorphism
s̄ s : Qn (A|A0 ) ∼
= Qn+2 (ΣA|ΣA0 ).
The relationship between σ and s̄ s remains unsettled.
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440
HANS-JOACHIM BAUES
(3.7) Proposition. Let A be a cogroup in space r / ', for example let A be a
suspension. Then the operator H 0 in (3.5) and H in (3.1) yield the following
commutative diagram
Qn+1 (ΣA)
& H0
H.
∼
=
Qn+1 (ΣA|ΣA) −→ Qn−1 (A|A)
σ
where σ is the isomorphism in (3.6).
Proof. Let µ : A → A ∨ A0 be the comultiplication of A with A = A0 . Then Σµ is
the comultiplication of the suspension ΣA. Now the naturality of H 0 in (3.5) shows
that the following diagram commutes
H0
Qn+1 (ΣA ∨ ΣA0 ) −−−−→ Qn−1 (A ∨ A0 |A ∨ A0 )
x
x

µ −(i ) −(i )
(Σµ)∗ −(i2 )∗ −(i1 )∗ 
 ∗ 2∗ 1∗
Qn+1 (ΣA)
H0
−−−−→
Qn−1 (A|A)
Here the vertical arrows map to the cross effects; in fact the left hand side is i12 H
by definition of H and the right hand side is i12 (1, T ) by the bilinearity of the cross
effect. This implies by the definition of σ 0 above that σ 0 H = (1, T )H 0 and hence
σH = H 0 .
For the metastable range of homotopy groups one has a diagram as in (3.7) where
σ is actually the double suspension up to sign; compare A.6.8 in [7] and example
(2.6) (B) above.
The next lemma justifies our choice of axioms of a quadratic homology theory.
It is the quadratic analogue of the uniqueness lemma in (2.7) above. The spherical
groups of a quadratic homology theory Q on pair r are the groups Qn (S r+k ) and
Qn (S r |S r ), n ∈ Z, k ≥ 0. Now the following uniqueness lemma holds.
(3.8) Lemma. Let φ : Q → Q0 be a natural transformation of quadratic homology
theories on pair r compatible with ∂ and δ. Moreover assume φ is an isomorphism
on spherical groups. Then φ is an isomorphism
φ : Qn (X, Y ) ∼
= Q0n (X, Y )
for all finite CW-pairs (X, Y ) ∈ pair r and n ∈ Z.
Proof. Since Qn ( | ) is a homology theory in each variable we see that φ induces an
isomorphism Qn (X|Z) ∼
= Q0n (X|Z) for all finite X, Z. Compare (2.7). This shows
that for k ≥ 0 one gets the isomorphism φ : Qn (X) ∼
= Q0n (X) for all spaces X which
r+k
are finite one point unions of spheres S
. By exactness it suffices to show that
φ induces an isomorphism φX : Qn (X) ∼
= Q0n (X) for all finite X in space r . We
proceed by induction on the dimension of X. If dim(X) = r then X is a finite one
point union of spheres S r . Now assume φX is an isomorphism for all finite X with
dim(X) < r + k and let dim(Y ) = r + k. Then we may assume that Y = CA ∪A X
where (X, A) is a pair with dim(X) < r + k and A is a finite one point union of
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QUADRATIC HOMOLOGY
441
spheres S r+k−1 . Hence φX and φA are isomorphisms and therefore the quadratic
excision sequence shows that also
φ : Qn+1 (Y, X) ∼
= Q0 (Y, X)
n+1
is an isomorphism. Hence ∂-exactness shows that φY is an isomorphism since φX
is one.
Now let (Q, ∂, δ) be a quadratic homology theory on space r and let n ≥ r, m ∈
Z. By (3.1) we obtain the quadratic Z-module
(3.9)
H
P
Qm {S n } = (Qm (S n ) −→ Qm (S n |S n ) −→ Qm (S n ))
which describes the canonical quadratic structure of the spherical groups above.
We now recall the following notation concerning quadratic Z-modules; compare [5].
H
P
(3.10) Definition. Let M = (Me −→ Mee −→ Me ) be a quadratic Z-module, i.e.
a pair of homomorphisms H and P with HP H = 2H and P HP = 2P . Then
M induces functors Ab → Ab which carry A to A ⊗ M, A ∗0 M and A ∗00 M
respectively. Here A ⊗ M is the abelian group with generators a ⊗ m, [a, b] ⊗ n for
a, b ∈ A, m ∈ Me , n ∈ Mee and relations
(a + b) ⊗ m = a ⊗ m + b ⊗ m + [a, b] ⊗ Hm,
[a, a] ⊗ n = a ⊗ P (n),
where a ⊗ m is linear in m and [a, b] ⊗ n is linear in a, b and n. Now let
d
q
0 → A1 −→ A0 −→ A → 0
be a short exact sequence in Ab where A0 is free abelian. Then we obtain homomorphisms
d
d
2
1
A1 ⊗ A1 ⊗ Mee −→
A1 ⊗ M ⊕ A1 ⊗ A0 ⊗ Mee −→
A0 ⊗ M
with d1 d2 = 0 as follows:
d1 (a ⊗ m) = (da) ⊗ m,
d1 ([a, a0 ] ⊗ n) = [da, a0 ] ⊗ n,
d1 (a ⊗ b ⊗ n) = [da, b] ⊗ n,
d2 (a ⊗ a0 ⊗ n) = −a ⊗ da0 ⊗ n + [a, da0 ] ⊗ n
for a, a0 ∈ A1 , b ∈ A0 , m ∈ Me , n ∈ Mee . One can check that cok (d1 ) = A ⊗ M .
Moreover we set
A ∗0 M = ker (d1 )/im (d2 ),
A ∗00 M = ker (d2 ).
These are the derived functors of the functor Ab → Ab which carries A to A ⊗ M .
An abelian group N ∈ Ab yields the quadratic Z-module N = (N → 0 → N ) with
Nee = 0. In this case A ⊗ N is the usual tensor product and A ∗ N = A ∗0 N is the
usual torsion product. Clearly A ∗00 N = 0.
We now consider the quadratic homology of a Moore space M (A, n). A Moore
space M (A, n), n ≥ 2, is a simply connected CW-space X with homology Hn (X, Z)
= A and H̃i (X, Z) = 0 otherwise. If A is a free abelian group then M (A, n) is a
one point union of spheres S n , in particular M (Z, n) = S n . Each homomorphism
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442
HANS-JOACHIM BAUES
ϕ : A → B in Ab admits a realization ϕ̄ : M (A, n) → M (B, n) with Hn (ϕ̄) =
ϕ. Here ϕ̄ is unique up to homotopy if A is free abelian. There is the natural
homomorphism
λ : A ⊗ Qm {S n } → Qm (M (A, n))
(3.11)
defined as follows. For a, b ∈ A = πn M (A, n), u ∈ Qm (S n ), v ∈ Qm (S n |S n ) we set
λ(a ⊗ u) = Qm (a)(u),
λ([a, b] ⊗ v) = P Qm (a|b)(v).
Let λ Qm (M (A, n)) be the cokernel of λ.
(3.12) Proposition. Assume that the quadratic homology (Q, ∂, δ) satisfies the
colimit axiom. Then λ above is an isomorphism in case A is a free abelian group.
Moreover for a general abelian group A ∈ Ab there is the natural exact sequence
(n ≥ 2, m ∈ Z):
0 → A ∗0 Qm {S n } −→λ Qm+1 (M (A, n)) −→ A ∗00 Qm−1 {S n }
e
∂
h
λ
−→ A ⊗ Qm {S n } −→ Qm (M (A, n)) →λ Qm (M (A, n)) → 0.
This result generalizes 9.3 in [5]. If Q does not satisfy the colimit axiom then
Proposition (3.12) is still true for finitely generated abelian groups.
Proof. If A is free abelian we see by (3.6) [5] or (6.4) below that λ is an isomorphism.
Now let A ∈ Ab and let
d : X = M (A1 , n) → Y = M (A0 , n)
be a map which realizes d in (3.10). Then the Moore space M (A, n) is the mapping
cone M (A, n) = CX ∪d Y . Therefore we get by (3.4) the following commutative
diagram in which the column and the row are exact.
Qm (X|X)
d2
Qm (X) ⊕ Qm (X|Y )
TTT
TTT d
TT1T
¯
d1
TTT
TTT
i
Qm+1 (M (A,n),Y )
Qm (Y )
Qm+1 (M (A,n))j
)
/
i
/
/
Qm (M (A,n)) →
O
δ
λ
Qm−1 (X|X)
A0 ⊗ Qm {S n }
q∗
/
A ⊗ Qm {S n }
d2
Qm−1 (X) ⊕ Qm−1 (X|Y )
The map q∗ is surjective and the kernel of q∗ is the image of d1 . Moreover using
the definition of d1 , d2 in (3.4) and (3.10) we get for d1 , d2 in the diagram
ker (d1 )/im (d2 ) = A ∗0 Qm {S n },
ker (d2 ) = A ∗00 Qm {S n }.
We now define the operators in (3.12) as follows. The inclusion e is induced by d¯1
where cok (i) = cok (λ) = im (j). The map h is the restriction of δ. The map ∂
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QUADRATIC HOMOLOGY
443
in the proposition is induced by ∂(δ)−1 and e and h in the proposition are derived
from i and j in the row of the diagram.
4. Homotopy functors
Following Goodwillie [17, 18, 19, 20, 21] we consider for r ≥ 0 functors D of the
form
D : space r → space
(4.1)
or
D : space r → spectra
We say that D is a homotopy functor if D carries weak equivalences to weak equivalences and preserves filtered colimits up to homotopy. That is, for each sequence
of cofibrations X0 X1 . . . in space r the induced map
hocolim (DXi ) → D(colim Xi )
is a weak equivalence. Here hocolim is the homotopy colimit. We say that D is
reduced if ∗ → D(∗) is a weak equivalence. We shall consider various examples of
such homotopy functors below.
(4.2) Definition. Let D be a homotopy functor as in (4.1). We define the Dhomology of a space X ∈ space r by the homotopy group
Hn (X; D) = πn (D(X)).
Moreover the relative D-homology is the relative homotopy group
Hn+1 (X, Y ; D) = πn+1 (D(X), D(Y ))
for (X, Y ) ∈ pair r . This yields the natural boundary map
∂ : Hn+1 (X, Y ; D) → Hn (Y ; D).
If D maps to spectra then all such homotopy groups are well defined abelian
groups for n ∈ Z. If D maps to space then these groups are abelian only for n ≥ 2.
For n = 1, resp. n = 0, the D-homology Hn (X, D) and Hn+1 (X, Y ; D) is a group,
resp. a pointed set. If D maps to space 1 we set Hn (X, D) = Hn (X, Y ; D) = 0 for
n ≤ 0. If D is a reduced homotopy functor we obtain as in (2.4) the suspension
s = q∗ ∂ −1 : Hn (X; D) → Hn+1 (ΣX; D)
which defines the stable theory HnS (X; D) of stable homology groups with coefficients
in D.
(4.3) Remark. We have for a reduced homotopy functor D the linearization Dlin
which is the homotopy functor obtained by the homotopy colimit of
D(X) → ΩD(ΣX) → Ω2 D(Σ2 X) → . . .
We clearly have the natural isomorphism
HnS (X, D) = Hn (X, Dlin ).
Goodwillie [17] showed that HnS (X; D) is a homology theory if D is approximately
1-excisive. This is also true if D is n-excisive for n ≥ 1 as follows from 3.2.4 in [20].
The following lemma is an easy consequence of the definitions.
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444
HANS-JOACHIM BAUES
(4.4) Lemma. Let D be a reduced homotopy functor which maps to space 2 or
spectra. Then D-homology is a boundary theory which satisfies the colimit axiom
in (2.1).
Proof. The ∂-exact sequence is otained by (2.6) (B). Moreover homotopic maps
f ' g induce f∗ = g∗ in D-homology since the projection I(X) → X of the
cylinder is a homotopy equivalence and hence a weak equivalence.
By an n-cube of spaces (or spectra) we will mean the following. Let P(n) be the
category whose objects are the subsets K of {1, 2, . . . , n} and whose morphisms
are the inclusion maps among the subsets. An n-cube X in a category C is a
covariant functor X : P(n) → C. Goodwillie defines and uses particular n-cubes
in space or spectra, namely Cartesian and co-Cartesian n-cubes. Let “holim”
be the homotopy inverse limit and “hocolim” be the homotopy colimit as defined
in Bousfield-Kan [12]. Let P 0 (n), resp. P 00 (n), be the full subcategory of P(n)
consisting of all K with K 6= φ, resp. K 6= {1, . . . , n}, and let X 0 and X 00 be the
restrictions of X to P0 (n), resp. P00 (n). There are maps
a(X) : X(φ) = lim(X) ' holim(X) → holim(X 0 ),
b(X) : hocolim(X 00 ) → hocolim(X) ' colim(X) = X({1, . . . , n}).
Now X is Cartesian, resp. co-Cartesian, if a(X), resp. b(X), is a weak equivalence.
An n-cube is strongly co-Cartesian if each of its 2-faces is co-Cartesian. A Cartesian
2-cube is also called a homotopy pull back and a co-Cartesian 2-cube is a homotopy
push out. A basic notion in [17, 18] is the following definition.
(4.5) Definition. Let D be a homotopy functor as in (4.1). Then D is termed nexcisive if D(X) is Cartesian for every strongly co-Cartesian (n + 1)-cube X in
space r . We say that D is linear if D is reduced and 1-excisive and we say that D
is quadratic if D is reduced and 2-excisive.
Goodwillie [17] proved the following result:
(4.6) Theorem. Let D be a homotopy functor as in (4.1) which maps to space 1
or spectra and let D be linear. Then D-homology is a homology theory.
We obtain the corresponding result for the quadratic case as follows.
(4.7) Theorem. Let D be a homotopy functor as in (4.1) which maps to space 2
or spectra and let D be quadratic. Then D-homology is a quadratic homology
theory on pair r in the sense of (3.2).
Here we use space 2 since we want all D-homology groups to be abelian.
Remark. Brown’s representability theorem (see 2.8) shows that the conclusion in
(4.6) admits a converse in the sense that each homology theory restricted to the
category of finite CW-complexes is the D-homology for an appropriate linear homotopy functor D. We do not know whether there is such a converse for theorem (4.7)
as well. One might need additional axioms to characterize the quadratic homology
theories obtained from quadratic homotopy functors.
Proof of (4.7). We obtain (3.2) (i) by (4.3) and (3.2) (ii) by (4.9) below; see also
(4.11). Moreover (3.2) (iii) is proved in (4.12) below.
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QUADRATIC HOMOLOGY
445
We need the following lemma; compare 1.18 [18]. Let X, Y be n-cubes in space
and let f : X → Y be a map between n-cubes which may be considered to be a
(n + 1)-cube. Let hofib(f ) be the n-cube obtained by taking the homotopy fibers
of f (K) : X(K) → Y (K), K ∈ P(n).
(4.8) Lemma. Assume Y (φ) is a connected space. Then f is a Cartesian (n + 1)cube if and only if hofib(f ) is a Cartesian n-cube.
Below we shall also consider quadratic homotopy functors D which map to
space 1 and not necessarily to space 2 as assumed in (4.7). For such homotopy
functors the following two lemmas hold.
(4.9) Lemma. Let D be a quadratic homotopy functor as in (4.1) which maps to
space 1 or spectra. Then D-homology defines for n ∈ Z a quadratic functor
Hn (−; D) : space r → Gr
which carries X to Hn (X, D). Moreover the suspension theory of cross effects
Hn (X|Y ; D) defined as in (2.11) is a homology theory.
The lemma is a consequence of Goodwillie’s results. In fact, let D(X ∨ Y )2 be
the homotopy fiber of D(r2 ) : D(X ∨ Y ) → D(Y ) and let D(X|Y ) be the homotopy
fiber of D(r1 ) : D(X ∨ Y )2 → D(X). Then one readily checks that one has
(4.10)
Hn (X|Y, D) = πn D(X|Y )
for any reduced homotopy functor D. If D maps to space 1 then D(X ∨ Y )2 is
always a connected space. We now repeat an argument of Goodwillie which proves
(4.9).
Proof of (4.9). We first show that the functor DX with DX (Y ) = D(X ∨ Y )2 is
1-excisive. In fact, let Y be a co-Cartesian 2-cube. Then r2 : X ∨ Y → Y is
a strongly co-Cartesian 3-cube; compare (A.1). Hence D(r2 ) = D(X ∨ Y → Y )
is a Cartesian 3-cube since D is 2-excisive. Therefore (4.8) shows that DX (Y )
is Cartesian and hence DX is 1-excisive. Now D(X|Y ) is the homotopy fiber of
DX (Y ) → DX (∗). Therefore the functor Y 7−→ D(X|Y ) is linear since DX is
1-excisive. This completes the proof of (4.9) by using (4.6).
(4.11) Remark. Let D be a quadratic homotopy functor as in (4.1) which maps to
space 1 . Goodwillie showed that there exists a spectrum E and a natural isomorphism
(1)
Hn (X|Y ; D) = πn (E ∧ X ∧ Y )
which is compatible with the suspension s of X and the suspension s̄ of Y ; see (2.11),
(2.12). Moreover the isomorphism is compatible with the interchange map on both
sides. Here the interchange map T on πn (E ∧ X ∧ Y ) is introduced by a Σ2 -action
t on E and by the interchange TX,Y : X ∧ Y ≈ Y ∧ X, that is T = πn (t ∧ TX,Y ). Let
(E ∧ X ∧ X)T be the homotopy orbit spectrum given by T . Then the Goodwillie
tower of the quadratic functor D yields a natural fibration sequence
(2)
Ω∞ (E ∧ X ∧ X)T → D(X) → Dlin (X).
Here the linearization satisfies Dlin (X) = Ω∞ (E 0 ∧ X) for an appropriate spectrum
E0.
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446
HANS-JOACHIM BAUES
(4.12) Lemma. Let D be a quadratic homotopy functor as in (4.1) which maps
to space1 or spectra. Then for Qn (X, Y ) = Hn (X, Y ; D) there exists a quadratic
excision sequence as in (3.2) (iii) which is natural and exact.
We point out that in the lemma Q1 (X, Y ) = H1 (X, Y ; D) is only a pointed set
if D maps to space 1 .
Proof of (4.12). Let (X, A) be an object in pair r and let f : A → X be the
inclusion. Then (X, A) yields a 3-cube in space r termed Cube (X, A) which is
obtained as the following map F between 2-cubes
1∨f
A ∨ X ←−−−− A ∨ A


(1,1)

(f,1)y
y
(1)
X
←−−−−
f
X ←−−−−

q
y
F
−−−−→
A


y
X/A ←−−−− ∗
A
f
Here F is defined by (0, 1) : A ∨ X → X, (0, 1) : A ∨ A → A, q : X → X/A and
0 : A → ∗. One readily checks that F = Cube (X, A) is well defined. Each square
in Cube (X, A) is a homotopy push out so that Cube (X, A) is actually strongly
co-Cartesian. Since D is quadratic this implies that D(Cube (X, A)) is Cartesian
and therefore by (4.8) the following diagram of homotopy fibers is a homotopy pull
back.
(1∨f )∗
D(A ∨ A)2 −−−−→ D(A ∨ X)2



(f,1)
(1,1)∗ y
∗
y
(2)
D(A)
f∗
−−−−→
P (q∗ )
Here P (q∗ ) is the homotopy fiber of q∗ : D(X) → D(X/A). Let K(A) be the
homotopy fiber of (1, 1)∗ : D(A ∨ A)2 → D(A). Then we get the fiber sequence
(f,1)∗
(1∨f )∗
(3)
K(A) −−−−→ D(A ∨ X)2 −−−−→ P (q∗ )
since (2) is a homotopy pull back. This fiber sequence of spaces or spectra induces
a long exact sequence of homotopy groups. The definition of K(A) yields for n ∈ Z
the following commutative diagram of short exact sequences of groups.
0 −−−−→
πn K(A)
k
−−−−→
πn D(A ∨ A)2
(1,1)∗
−−−−→ πn D(A) −−−−→ 0
k
k
(1,P )
0 −−−−→ kernel(1, P ) −−−−→ πn D(A) ⊕ πn D(A|A) −−−−→ πn D(A) −−−−→ 0
Here we use the isomorphism in (4.10) and (2.4) (4) and we point out that for
D : space r → space 1 the fiber D(A ∨ X)2 is connected and that π1 (D(A|A)) is
central and split in π1 D(A ∨ X)2 . Now an isomorphism of abelian groups
(4)
Θ : πn D(A|A) ∼
= kernel(1, P ) = πn K(A)
is obtained by mapping y to (P y, −y).
The boundary operator ∂ of the long exact sequence of homotopy groups associated to (3) yields the natural transformation
(5)
∂
δ : Hn+1 (CA ∪A X, X; D) = πn P (q∗ ) −→ πn−1 K(A) = Hn−1 (A|A; D).
This completes the proof of the quadratic excision axiom.
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QUADRATIC HOMOLOGY
447
We can use (4.12) for the proof of the following result.
(4.13) Proposition. Let D : space 1 → space 1 be a quadratic homotopy functor.
Then we obtain by (3.1) and (4.9) the square group
H1 {S 1 ; D} = (H1 (S 1 ; D) −→ H1 (S 1 |S 1 ; D) −→ H1 (S 1 , D))
H
P
and using the tensor product in (6.3) below there is a natural isomorphism
H1 (Y, D) ∼
= π1 (Y ) ⊗ H1 {S 1 ; D}
for 2-dimensional CW-complexes Y .
Proof. We may assume that Y = CA ∪A X where A and X have the homotopy
type of one point union of 1-spheres. We now apply (4.12) and ∂-exactness. This
gives us the exact sequence of groups, Q1 (X) = H1 (X, D),
Q1 (A ∨ X)2 → Q1 (X) → Q1 (Y ) → 0.
Since D preserves filtered colimits up to homotopy we get by [10] with M =
H1 {S 1 ; D}:
Q1 (X) = π1 (X) ⊗ M,
Q1 (A ∨ X)2 = Q1 (A) ⊕ Q1 (A|X)
= π1 (A) ⊗ M ⊕ π1 (A)ab ⊗ π1 (X)ab ⊗ Mee
d
and therefore the result follows from (8.4) [10]. For this observe that π1 (A) →
q
π1 (X) → π1 (Y ) has the property that q is surjective and the normal closure of
image of d is the kernel of q.
(4.14) Example. It follows from Goodwillie’s Calculus [19, 20] that there is a sequence of functors Pn from pointed spaces to pointed spaces and natural transformations
X 8RRR
/
Pn (X)
88 RRRR
RRR
88
RRR
88
RRR
88
88
Pn−1 (X)
88
88
88
88
88
..
88
.
88
88
88
(
P1 (X) = Q(X) = Ω∞ Σ∞ (X)
Here Pn are homotopy functors satisfying n-th order excision and the maps X →
Pn (X) are (n + 1)k + 1 connected, where k is the connectivity of X. The functors
Pn are uniquely determined by these universal properties. According to Goodwillie
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448
HANS-JOACHIM BAUES
the tower above is termed the Taylor tower of the identity. Here P1 is a linear
homotopy functor and the homology
Hn (X, P1 ) = πnS (X)
coincides with stable homotopy groups. Therefore the quadratic homotopy functor
P2 yields a canonical quadratic homology which we call quadratic homotopy groups:
Hn (X, P2 ) = πnQ (X).
This is the quadratic analogue of stable homotopy groups. We know by (4.7)
that quadratic homotopy groups πnQ form a quadratic homology theory on space 2
satisfying all properties in section 3. The cross effect of πnQ (X) is given by a natural
isomorphism
S
(X ∧ Y )
πnQ (X|Y ) = πn+1
where X ∧ Y = X × Y /X ∨ Y , compare [19], [24]. Arone-Mahowald [1] mentioned
that P2 (X) also can be constructed by the fiber of the stable James-Hopf map γ2
in [13]; that is
γ2
P2 (X) → Q(X) −→ Q(X ∧ X)T
is a natural fiber sequence. Here (X ∧ X)T is the homotopy orbit space of the
Z/2-action on X ∧ X given by the interchange map TX,X . In the metastable range
Q
{S n } in table 2 in
we have considered examples of the quadratic Z-modules πn+k
9.9 [5]. Such quadratic Z-modules seem to be the appropriate quadratic analogue
of stable homotopy groups of spheres.
5. Homotopy functors induced by endofunctors
of the category of groups
Let Gr be the category of groups and let sGr be the category of simplicial
groups. A functor F : Gr → Gr induces the functor
F̄ : sGr → sGr
which carries the simplicial group X to the simplicial group F ◦X. We now consider
the following composition of functors F] = β F̄ α:
(5.1)
α
β
F̄
space 1 −→ sGr −→ sGr −→ space 1 .
Here we obtain α as follows. For a pointed space X let S(X) be the reduced
singular set consisting of all singular simplexes σ : ∆n → X with σ(v) = ∗ for all
vertices v of the simplex ∆n . Then the functor α carries X to the Kan-loop group
α(X) = GS(X) of S(X); see for example [15]. Moreover for a simplical group
G let |G| be the realization and let β(G) = B|G| be the classifying space of the
topological group |G|. Then α and β induce equivalences of homotopy categories
α
space 1 / ' Ho(sGr)
β
where β is the inverse of α. Here Ho(sGr) is the localization of sGr with respect to
weak equivalences. Let gr be the full subcategory of Gr consisting of free groups.
Since α(X) above is actually a free simplicial group we see that F] depends only on
the restriction F0 : gr → Gr of F . On the other hand each functor F0 : gr → Gr
determines a unique extension F1 : Gr → Gr with the property that F1 preserves
cokernels. Therefore we may assume that F in (5.1) preserves cokernels.
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QUADRATIC HOMOLOGY
449
Moreover it is convenient to assume that the behaviour of F] on infinite CWcomplexes is determined by its behaviour on finite complexes. Therefore we assume
that F] preserves filtered colimits up to homotopy. One readily checks that this is
the case if and only if F preserves filtered colimits. These remarks lead to the
(5.3) Definition. A group functor is an endofunctor F : Gr → Gr of the category
of groups which preserves coequalizers and filtered colimits. Hence such a group
functor is determined by its restriction to the full subcategory of finitely generated
free groups.
(5.4) Lemma. A group functor F : Gr → Gr induces a functor F] : space 1 →
space 1 which is a reduced homotopy functor.
We point out that the lemma does not imply that the functor F̄ in (5.1) carries
weak equivalences in sGr to weak equivalences in sGr; this in general does not
hold.
Proof of (5.4). Clearly ∗ → F] (∗) is a homotopy equivalence so that F] is reduced.
We have to show that F] carries homotopy equivalences to weak equivalences. Let
H : f0 ' f1 be a homotopy in space 1 . Then there exists a homotopy H 0 :
αf0 ' αf1 since α(X) is a free simplicial group. The functor F̄ carries H 0 to a
homotopy F̄ αf0 ' F̄ αf1 in the category of simplicial sets, see (1.10) and (4.2) in
[15]. This implies that πn (F̄ αf0 ) = πn (F̄ αf1 ) and hence πn (F] f0 ) = πn (F] f1 ) for
n ∈ Z. From this one readily derives that F] carries homotopy equivalences to weak
equivalences.
(5.5) Definition. Let F : Gr → Gr be a group functor and let X ∈ space 1 . We
define the F -homology of X,
Hn (X; F ) = Hn (X; F] ) = πn (F] X),
by the F] -homology of X, n ∈ Z. Similarly we obtain the relative F -homology
Hn (X, Y ; F ) of a pair (X, Y ) ∈ pair 1 . Clearly F -homology has the properties of
D-homology described in (4.2) and (4.4). In particular H1 (X; F ) is only a group;
see (5.11). Moreover we obtain the stable F -homology
HnS (X, F ) = Hn (X, F]lin ).
Compare (4.3). Let F, G : Gr → Gr be group functors. Then a natural transformation t : F → G induces natural transformations
t] : F] (X) → G] (X) and
t∗ : Hn (X, Y ; F ) → Hn (X, Y ; G)
where t∗ is the coefficient homomorphism induced by t] . We obtain t] as follows.
Let t̄ : F̄ → Ḡ be the transformation induced by F . Then we set t] = β t̄αX .
Clearly the coefficient homomorphism t∗ is compatible with the boundary map ∂
of F -homology.
(5.6) Lemma. Let (X, Y ) ∈ pair 1 be an r-connected pair. Then Hn (X, Y ; F ) = 0
for n ≤ r. This implies that
Hn (X; F ) = Hn (X n+1 ; F )
depends only on the (n + 1)-skeleton of X. Moreover if X is r-connected then so is
F] (X).
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450
HANS-JOACHIM BAUES
Proof. We may assume that X is a reduced CW-complex with subcomplex Y and
that X − Y has only cells in dimension > r. Recall the notion of a “free simplicial
group” as defined in section 4 of [15]. By a result of Kan [25] we obtain a homotopy
equivalence
ϕ : (GX , GY ) ' (αX, αY )
of pairs of free simplicial groups. Here GX as a free simplicial group has generators
in degree t which are exactly the (t + 1)-cells of X, t ≥ 0. Moreover GY is the
subobject generated by the cells of Y . Hence (GX )n = (GY )n for n < r and
therefore also (F̄ GX )n = (F̄ GY )n for n < r. Since F̄ carries ϕ above to a weak
equivalence we see that Hn (X, Y ; F ) = 0 for n ≤ r.
Let I be the identity functor of the category Gr. Then one has the canonical
natural isomorphism
(5.7)
πn (X, Y ) = Hn (X, Y ; I)
which we use as an identification. For a group G let Γk (G) ⊂ G be the subgroup of
k-fold commutators. Then G/Γ2 G = ab(G) = Gab is the abelianization of G. For
k ≥ 1 we obtain the nilization functors
(5.8)
nilk : Gr → Gr
which carry G to the quotient nilk (G) = G/Γk+1 G. Hence nil1 = ab is the abelianization functor. All functors nilk , k ≥ 1, are group functors in the sense of (5.3).
By the result of Dold-Kan it is well known that for (X, Y ) ∈ pair 1 one has the
natural isomorphism
(5.9)
Hn (X, Y ; ab) = Hn (X, Y )
where the right hand side is the singular homology. Moreover the natural transformation I → ab given by the quotient map G → ab(G) induces the classical Hurewicz
homomoprhism
πn (X, Y ) = Hn (X, Y ; I) → Hn (X, Y ; ab) = Hn (X, Y ).
Similarly one obtains by the natural quotient map I → nilk (G), the nilk -Hurewicz
homomorphism
πn (X, Y ) = Hn (X, Y ; I) → Hn (X, Y ; nilk ).
The following generalization of the classical Hurewicz theorem is due to Curtis [15].
(5.10) Curtis theorem. Let r ≥ 2 and let X be an (r − 1)-connected space. Then
the nilk -Hurewicz homomorphism
πn (X) → Hn (X; nilk )
is an isomorphism for n < r+{log2 (k+1)} and is surjective for n = r+{log2 (k+1)}.
Here {a} denotes the least integer ≥ a.
The Hurewicz theorem is the case k = 1 of this result since {log2 (2)} = {1} = 1.
For k = 2 we have {log2 (3)} = 2 since 2 > log2 (3) > 1. Hence the nil2 -Hurewicz
homomorphism yields for an (r − 1)-connected space X, r ≥ 2,
πr (X) = Hr (X, nil2 ),
πr+1 (X) = Hr+1 (X, nil2 ),
(5.11)
πr+2 (X) Hr+2 (X, nil2 ),
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QUADRATIC HOMOLOGY
451
where denotes a surjection. Clearly the functor nil1 = ab is linear and the
functor nil2 is quadratic in the sense of (2.9). In the next section we consider all
quadratic group functors Gr → Gr. It would be interesting to understand the
stable theory HnS (X, nilk ), k ≥ 1, which is a homology theory approximating for
k → ∞ stable homotopy πnS (X). We shall determine HnS (X, F ) for any quadratic
group functor in (8.5). This in particular yields an explicit spectrum E for which
HnS (X, nil2 ) = Hn (X, E); compare (8.16).
The next result corresponds to (4.13) in case F is a quadratic group functor.
(5.12) Lemma. Let F : Gr → Gr be a group functor. Then we have for X ∈
space 1 the natural isomorphism
H1 (X, F ) = F (π1 (X)).
Moreover if S is a one point union of 1-spheres we have
Hn (S, F ) = 0
for
n ≥ 2.
Proof. The group G = π1 (S) is a free group for which the constant simplicial group
Ḡ with Ḡn = G for n ≥ 0 is a free simplicial group which is homotopy equivalent
to α(S). Hence one has a weak equivalence F̄ (Ḡ) → F̄ α(S) where F̄ (Ḡ) is again
a constant simplicial group. This shows Hn (S, F ) = πn−1 F̄ (Ḡ) = 0 for n ≥ 2 and
H1 (S, F ) = π1 F̄ (Ḡ) = F (G). Now let H = αX with π0 H = π1 X. Then the degree
1 part of the simplicial group H
H1 ⇒ H0 → π0 H
is a coequalizer. Since F preserves cokernels also
F H1 ⇒ F H0 → F π0 H
is a coequalizer and therefore π0 F H = F π0 H. This shows H1 (X, F ) = F (π1 X).
If F = ab is the abelianization functor we have by (5.9) and (5.11)
H1 (X) = H1 (X, ab) = ab(π1 (X)).
This is a part of the classical Hurewicz theorem.
6. Square-homology
We show that the homology theories obtained by linear group functors Gr → Gr
are exactly the classical “ordinary homology theories” of Eilenberg-Mac Lane. This
motivates the study of homology defined by quadratic group functors Gr → Gr.
There is the following classification of linear functors in [10].
(6.1) Lemma. The category of linear group functors Gr → Gr is equivalent to
the category Ab of abelian groups. More precisely for each linear group functor F
there is an abelian group A and an isomorphism
F (G) = ab(G) ⊗ A
which is natural in G ∈ Gr. The equivalence carries F to A = F (Z). In particular
F admits a factorization
ab
⊗A
F : Gr −→ Ab −→ Ab ⊂ Gr.
As a consequence of this result and the Dold-Kan theorem [16] we get
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452
HANS-JOACHIM BAUES
(6.2) Proposition. Let F be a linear group functor corresponding to A ∈ Ab as
in (6.1). Then one has for (X, Y ) ∈ pair 1 the natural isomorphism, n ∈ Z,
Hn (X, Y ; F ) = Hn (X, Y ; A)
where the right hand side is the singular homology with coefficients in A.
This implies that the homology theories of linear group functors are exactly the
ordinary homology theories in the sense of Eilenberg-Steenrod which satisfy the
classical ”dimension axiom”. As a next step we consider quadratic group functors.
For this recall from [10] the following notation on square groups.
(6.3) Definition. A square group
H
P
M = (Me −→ Mee −→ Me )
is given by a group Me and an abelian group Mee . Both groups are written additively. Moreover P is a homomorphism and H is a quadratic function, that is the
cross effect
(a|b)H = H(a + b) − H(b) − H(a)
is linear in a, b ∈ Me . In addition the following properties are satisfied (x, y ∈ Mee ).
(1)
(P x|b)H = 0 and (a|P y)H = 0,
(2)
P (a|b)H = a + b − a − b,
(3)
P HP (x) = P (x) + P (x),
By (1) and (2) P maps to the center of Me and by (2) the cokernel of P is abelian.
Hence Me is a group of nilpotency degree 2. Let Square be the category of square
groups. As an example we have the square group
H
0
Znil = (Z −→ Z −→ Z)
r
and P = 0; many other examples are discussed in [9, 10]. A
2
quadratic Z-module M is a square group for which H is linear and HP H = 2H;
see (3.1).
with H(r) =
Let G be a group and let M be a square group. Similarly as in (3.10) we define
the group G ⊗ M by the generators g ⊗ a and [g, h] ⊗ x with g, h ∈ G, a ∈ Me and
x ∈ Mee subject to the relations
(g + h) ⊗ a = g ⊗ a + h ⊗ a + [g, h] ⊗ H(a),
[g, g] ⊗ x = g ⊗ P (x)
where g ⊗ a is linear in a and where [g, h] ⊗ x is central and linear in each variable
g, h and x. There are obvious induced maps for this tensor product so that one
gets a bifunctor
⊗ : Gr × Square → Gr.
One has the natural isomorphism
nil2 (G) = G ⊗ Znil .
In [5, 10] we obtain the following classification of quadratic group functors which
is the quadratic analog of (6.1). Let Nil be the category of groups of nilpotency
degree 2.
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QUADRATIC HOMOLOGY
453
(6.4) Proposition. The category of quadratic group functors Gr → Gr is equivalent to the category Square of square groups. More precisely for each quadratic
group functor F one has the square group M = F {Z} in (3.1) and isomorphisms
F (G) = G ⊗ M = nil2 (G) ⊗ M
which are natural in G ∈ Gr. The equivalence carries F to M and F = nil2 corresponds to M = Znil . In particular quadratic group functors admit a factorization
nil
⊗M
Gr −→2 Nil −→ Nil ⊂ Gr.
Moreover the quadratic group functors which admit a factorization
ab
Gr −→ Ab −→ Ab ⊂ Gr
correspond exactly to quadratic Z-modules M .
Using this result we identify square groups M and quadratic group functors
Gr → Gr; we write M : Gr → Gr for the functor which carries G to G ⊗ M . As a
quadratic analog of (6.2) we define for a square group M and (X, Y ) ∈ pair 1 the
square-homology with coefficients in M , n ∈ Z,
(6.5)
Hn (X, Y ; M ) = Hn (X, Y ; M] ) = πn (M] X, M] Y ).
Here M] is the homotopy functor (5.1) induced by M . The groups Hn (X; M )
and Hn+1 (X, Y ; M ) are abelian for n ≥ 2 and of nilpotency degree 2 for n = 1.
Moreover H1 (X, Y ; M ) is a pointed set. For n ≤ 0 all groups (6.5) are trivial.
(6.6) Theorem. Let F : Gr → Gr be a quadratic group functor. Then F] in (5.1)
is a quadratic homotopy functor.
We prove this result in the Appendix A. Since for a simply connected space X
F] X is also simply connected and we obtain by (6.6), (6.4), (4.7) the corollary:
(6.7) Corollary. Square-homology Hn (X, Y ; M ) with coefficients in a square group
M is a quadratic homology theory on pair 2 in the sense of (3.2).
Using (4.3) we derive from (6.6) the next corollary which yields the homology
theory or spectrum associated to a square group.
(6.8) Corollary. Stable square-homology HnS (X, Y ; M ) with coefficients in a
square group M is a homology theory on pair 1 .
In Corollary (6.7) we use the category space 2 in order to obtain abelian groups
Hn (X, Y ; M ). The next result improves (6.7) for the category space 1 .
(6.9) Theorem. Let M be a square group. Then square-homology with coefficients
in M defines a quadratic functor
Ab for n ≥ 2,
Hn (−; M ) : space 1 →
Nil for n = 1
with cross effect
Hn (X|Y ; M ) = Hn+1 (X ∧ Y ; Mee )
for n ∈ Z. Here the right hand side is the singular homology of the smash product
X ∧ Y with coefficients in the abelian group Mee . The isomorphism is compatible
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454
HANS-JOACHIM BAUES
up to sign with the suspension s and s̄ and the interchange map on both sides.
Moreover for (X, A) ∈ pair 1 one has the natural exact sequence of groups:
Hn−1 (A; M ) ⊕ Hn (A ∧ X; Mee )
k
j]
δ
−→ Hn (A ∧ A; Mee ) −→ Hn (CA ∨ X, A ∨ X; M )
i]
δ
−→ Hn (CA ∪A X, X; M ) −→ Hn−1 (A ∧ A; Mee )
The operators i] and j] are defined as in (3.2) (iii). For n ≤ 1 the sequence consists
of trivial groups. As in (3.5) we derive from the quadratic excision sequence in (6.9)
the following long exact sequence.
(6.10) Corollary. Let M be a square group and A ∈ space 1 . Then one has the
natural long exact sequence of groups, n ∈ Z,
P
H0
s
P
−→ Hn (A; M ) −→ Hn+1 (ΣA; M ) −→ Hn (A ∧ A; Mee ) −→ Hn−1 (A; M ) → .
Proof of (6.9). In the following proof we do not use (6.6). Let F : Gr → Gr
be given by F (G) = G ⊗ M where M is the square group. By [10] we have for
X, Y ∈ Gr the binatural isomorphism
F (X|Y ) = ab(X) ⊗ ab(Y ) ⊗ Mee .
(1)
We now consider the following commutative cubical diagram in Gr which is determined by the inclusion
f :A→A∨C =X
(2)
where X is the sum of A and C in Gr. The pair (X, A) yields the three cube
G = Cube(X, A) as in (4.12):
1∨f
(3)
f
A ∨ X ←−−−− A ∨ A


(1,1)

(f,1)y
y
X
←−−−−
G
−−−−→
X ←−−−−

q
y
A


y
X/A ←−−−− ∗
A
f
Here the map G between 2-cubes is given by (0, 1) : A ∨ X → X, (0, 1) : A ∨ A → A,
q = (0, 1) : A ∨ C = X → X/A = C and 0 : A → ∗. One readily checks that
Cube (X, A) is well defined. We now apply F to Cube (X, A) and we obtain the
following square consisting of the kernels of all F (g) where g is one of the arrows
in G.
(1∨f )∗
(4)
F (A ∨ A)2 −−−−→ F (A ∨ X)2


(1,1)

(1,1)∗ y
∗
y
F (A)
−−−−→ kernel(F q)
f∗
Using the assumption that X = A ∨ C we see by (1) that (1 ∨ f )∗ in (4) induces an
isomorphism of the kernels of the vertical arrows in (4). Hence (4) is a pull back
diagram in Gr. Now let
(5)
f :A→X
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QUADRATIC HOMOLOGY
455
be a cofibration of simplicial groups. Then we know for all n that we can choose
free groups Cn such that
Xn = An ∨ Cn .
(6)
Hence f in (4) is of the form (2) for each n. This implies by the naturality of (3)
and (4) that also (4) with f as in (5) is a pull back diagram in the category of
simplicial groups. This yields the short exact sequence of simplicial groups
(f,1)∗
(1∨f )∗
(7)
0 → K(A) −→ F (A ∨ X)2 −→ kernel(F q) → 0,
(8)
K(A) = kernel{(1, 1)∗ : F (A ∨ A)2 → F (A)}.
Using F] in (5.1) we see that Hn (X 0 ; M ) = πn−1 (F X) where X = αX 0 is the
simplicial group associated to X 0 ∈ space r . Hence we get by (1) the cross effect
formula
Hn (X 0 |Y 0 ; M ) = πn−1 (F (X|Y ))
(9)
= πn−1 (ab(X) ⊗ ab(Y ) ⊗ Mee )
= Hn+1 (X 0 ∧ Y 0 ; Mee ).
Here the last isomorphism is given by the Eilenberg-Zilber theorem. This shows that
the cross effect formula in (6.9) is satisfied. It only remains to check the quadratic
excision sequence. Now a pair (X 0 , A0 ) in pair1 , corresponds to a cofibration A X in the category of simplicial groups for which we have the short exact sequence
(7). This sequence induces a long exact sequence of homotopy groups which is
isomorphic to the quadratic excision sequence for (X 0 , A0 ). The boundary ∂ for (7)
determines the natural operator
δ : Hn+1 (CA0 ∪ X 0 ; M ) = πn−1 kernel(F q) −→ πn−2 K(A)
∂
(10)
Θ
= πn−2 F (A|A) = Hn−1 (A0 |A0 ; M ).
Here the definition of K(A) in (8) yields the split short exact sequence
0 −−−−→
πn K(A)
k
−−−−→
πn F (A ∨ A)2
(1,1)∗
−−−−→ πn F (A) −−−−→ 0
k
k
(1,P )
0 −−−−→ kernel(1, P ) −−−−→ πn (F A) ⊕ πn F (A|A) −−−−→ πn F (A) −−−−→ 0
where we use the isomorphism in (2.9) (4). Now the isomorphism Θ in (10)
(11)
Θ : πn (F (A|A)) ∼
= kernel(1, P ) = πn K(A)
is obtained by mapping y to (P y, −y). This corresponds to the definition of j] in
(3.2) (iii).
The first nonvanishing homology with coefficients in a square group is described
by the following result.
(6.11) Proposition. Let M be a square group and let M add = cok(P ) ∈ Ab be the
cokernel of P : Mee → Me . If X is an (r − 1)-connected space one has
(
π1 (X) ⊗ M for r = 1,
Hr (X, M ) =
πr (X) ⊗ M add for r > 1.
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456
HANS-JOACHIM BAUES
Proof. For r = 1 this is a consequence of (5.11) or (4.13). Now let r ≥ 2. The
EHP-sequence shows that for a one point union S of 1-spheres the sequence
P
ab(G) ⊗ ab(G) ⊗ Mee −→ G ⊗ M → H2 (ΣS; M ) → 0
is exact where G = π1 (S). This implies
H2 (ΣS; M ) = ab(G) ⊗ M add .
Moreover the EHP-sequence shows that H2 (ΣS; M ) = Hn (Σn−1 S; M ) for n ≥ 2.
Now we have
Hr (X; M ) = Hr (X r+1 ; M ).
Here X r+1 is the mapping cone of a maps f : S 0 → S 00 where S 0 and S 00 are one
point unions of r-spheres. Now the ∂-exactness and the quadratic excision shows
Hr (X r+1 ; M ) = πr (X) ⊗ M add .
7. Square homology with coefficients
Let sAb be the category of simplicial abelian groups and let Chain be the
category of chain complexes C of abelian groups with Ci = 0 for i < 0. It is a
result of Dold and Kan that there are isomorphisms of categories (see [9])
N
(7.1)
sAb Chain
K
where K is the inverse of the normalization N . Here N (A) is also termed the Moore
chain complex of A. Kan [26], 15.1, showed that for a reduced simplicial set X and
its Kan loop group G(X) one has the isomorphism of chain complexes
(7.2)
N (A(X)) ∼
= s−1 C̃∗ X.
Here A(X) = ab G(X) is the abelianization of G(X) and C̃∗ X is the reduced
(normalized) chain complex of the simplicial set X. Recall that the suspension sk C
with k ∈ Z is (sk C)n = Cn−k with the differential d(sk x) = (−1)k sk (dx). We
define for X ∈ space1 the singular chain complex
(7.3)
C̃∗ X = C̃∗ (SX)
where SX is the reduced singular set in (5.1). Given a quadratic Z-module M we
obtain the induced chain functor M] which is the composite
K
⊗M
N
M] : Chain −→ sAb −→ sAb −→ Chain.
The next result is a consequence of (7.1), (7.2) and (6.4).
(7.4) Proposition. Let M be a quadratic Z-module and X ∈ space1 . Then the
square homology H∗ (X, M ) is determined by C̃∗ X. More precisely there is a natural
isomorphism, n ∈ Z,
Hn (X, M ) = Hn−1 (M] (s−1 C̃∗ X)).
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QUADRATIC HOMOLOGY
457
Proof. Using the definitions we get
Hn (X, M ) = πn (M] X)
= πn−1 ((GSX) ⊗ M )
= πn−1 ((GSX)ab ⊗ M )
(*)
= πn−1 ((KN (GSX)ab ) ⊗ M )
= πn−1 ((K s−1 C̃∗ X) ⊗ M )
= Hn−1 N ((K s−1 C̃∗ X) ⊗ M )
= Hn−1 M] (s−1 C̃∗ X).
In (*) we use the fact that for a group G we have G ⊗ M = Gab ⊗ M if M is a
quadratic Z-module; compare (6.4).
(7.5) Remark. The homotopy type of the chain complex C̃∗ X is determined by the
homology H∗ (X, Z). This shows by (7.4) that the abelian groups Hn (X, M ) are
determined by H∗ (X, Z) and M provided M is a quadratic Z-module. This is not
true if M is a square group. For example we obtain by (5.11)
0 = H3 (CP2 , Znil ) 6= H3 (S 2 ∨ S 4 , Znil ) = Z
with H∗ (CP2 , Z) = H∗ (S 2 ∨ S 4 , Z). Here CP2 is the complex projective plane and
S 2 ∨ S 4 is the one point union of spheres.
(7.6) Remark. The universal coefficient formula in [11] can be used to compute
H∗ (X, M ) if M is a quadratic Z-module. For example if Mee is torsion free one has
the natural short exact sequence
0 → (H̄∗ ⊗ M )n−1 → Hn (X, M ) → (H̄∗ ∗0 M )n−2 → 0
where H̄∗ = s−1 H̃∗ (X, Z) is the desuspended reduced homology of X. Here we use
the graded tensor and torsion products defined in [11].
Now let M be a square group. Then there is the canonical short exact sequence
q
i
0 → M̂ −→ M −→ M add → 0
(7.7)
in the category Square where M add = cok (P ) is an abelian group and where M̂
is a quadratic Z-module. More precisely we obtain the commutative diagram
H
P
H
P
M̂ = (im (P ) −−−−→ Mee −−−−→ im (P ))


∩
k
∩
y
M=


y
(Me


y
−−−−→ Mee −−−−→


y
M add =(cok (P ) −−−−→
0
Me )

q
y
−−−−→ cok (P ))
where q is the quotient map. The definition of a square group shows readily that
the restriction of H in M to M̂e = im (P ) is a homomorphism and that M̂ is a
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458
HANS-JOACHIM BAUES
quadratic Z-module. If G is a free group then (7.7) induces the short exact sequence
of groups
0 −−−−→ G ⊗ M̂ −−−−→ G ⊗ M −−−−→ G ⊗ M add −−−−→ 0
(1)
k
k
A ⊗ M̂
A ⊗ M add
where A = ab (G). If G is a free simplicial group this sequence is still well defined
and short exact. Hence the long exact sequence of homotopy groups for G = G(SX)
yields the following long exact sequence of square homology groups
(2)
∂
q∗
i
∂
∗
−→ Hn (X, M̂ ) −→
Hn (X, M ) −→ Hn (X, M add ) −→ Hn−1 (X, M̂ ) −→ .
Here H∗ (X, M add ) is the usual homology of X with coefficients in the abelian group
M add = cok (P : Mee → Me ). Moreover H∗ (X, M̂ ) depends by (7.5) only on the
homology H∗ (X, Z) and M̂ . In fact H∗ (X, M̂ ) can be computed by the universal
coefficient theorem [11] as follows; see (7.6).
Given a square group M we obtain the involution
T = HP − 1 : Mee → Mee
with T T = 1. This yields the chain complex of groups
M∗e = (Me ←− Mee ←− Mee ←− Mee ←− . . . )
P
1−T
1+T
1−T
where Me is in degree 0. We have 1 − T = 2 − HP and 1 + T = HP so that the
homology of M∗e is

cok (P ),
n = 0,



ker (P )/im (2 − HP ),
n = 1,
(7.8)
Hn (M∗e ) =

ker (2 − HP )/im (HP ),
n = 2k ≥ 2,



ker (HP )/im (2 − HP ),
n = 2k + 1 ≥ 3.
Moreover we associate with M the following quadratic Z-modules Zn M∗ , n ≥ 1.

j
2−HP

ker (P ) −→ Mee −→ ker (P ),
n = 1,



j
HP
(7.9) Zn M∗ = ker (2 − HP ) −→
Mee −→ ker (2 − HP ),
n = 2k ≥ 2,



j

2−HP
ker (HP ) −→ Mee −→ ker (HP ),
n = 2k + 1 ≥ 3.
Here j denotes the inclusion. We now compute the square-homology of a Moore
space M (A, n) of an abelian group A.
(7.10) Proposition. Let n ≥ 2 and let M be a square group. Then the square
homology of the Moore space M (A, n), A ∈ Ab, has the following properties. If A
is free abelian there is a natural isomorphism


0 for k < 0 or k ≥ n,
e
Hn+k (M (A, n); M ) = A ⊗ Hk M∗ for 0 ≤ k < n − 1,

A ⊗ Z
M for k = n − 1.
n−1
∗
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QUADRATIC HOMOLOGY
459
In the general case one gets for A ∈ Ab the following natural isomorphisms and
short exact sequences respectively.

0
for k < 0, k > n + 1,




A ⊗ cokP,
for k = 0,
Hn+k (M (A, n); M ) =
0

A
∗
Z
M
for k = n,
n−1 ∗


 00
A ∗ Zn−1 M∗ for k = n + 1,
0 → A ⊗ Zn−1 M∗ → H2n−1 (M (A, n); M ) → A ∗ Hn−2 M∗e → 0,
0 → A ⊗ Hr M∗e → Hn+r (M (A, n); M ) → A ∗ Hr−1 M∗e → 0
for 0 < r < n − 1.
In (8.4) we shall see that the short exact sequence for Hn+r (M (A, n); M ), 0 <
r < n − 1, is actually naturally split.
(7.11) Remark. We now determine the spherical groups of square homology; see
(3.8). Let M be a square group. For each sphere S n we obtain by (3.1) the square
group (n ≥ 1, k ∈ Z)
H
P
Hn+k {S n ; M } = (Hn+k (S n ; M ) −→ Hn+k (S n |S n ; M ) −→ Hn+k (S n ; M )).
For k ≥ 1 this is a quadratic Z-module as in (3.9). Using (5.11) we see
M, k = 0,
H1+k {S 1 ; M } =
0, otherwise.
Moreover by (7.10) we get for n > 1


0 for k < 0 or k > n − 1,
e
n
Hn+k {S ; M } = Hk M∗ for 0 ≤ k < n − 1,

Z
n−1 M∗ for k = n − 1.
Hence only H2n−1 {S n ; M } is quadratic and Hn+k {S n ; M } is an abelian group for
k 6= n − 1.
Proof of (7.10). For k = 0 compare (6.11). For k > 0 we have by the exact sequence
(7.7) the formula, X = M (A, n),
Hn+k (X; M ) = Hn+k (X, M̂ ).
If A is free abelian the right hand side is computed in [11] by
Hn+k (X, M̂ ) = (H̄∗ ⊗ M̂ )n+k−1 .
Here H̄∗ = s−1 H∗ (X, Z) is concentrated in degree n − 1 so that the result in [11]
yields the formulas of the proposition if A is free abelian. If A is not free abelian
we use the exact sequence in (3.12).
(7.12) Remark. If M is a quadratic Z-module we can use (7.10) for the computation
of square homology with coefficients in M . For this let X ∈ space1 and let
(1)
Xn = M (An , n) with
An = Hn (X, Z)
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460
HANS-JOACHIM BAUES
be the Moore space of the nth homology of X. Then there exists a homotopy
equivalence of chain complexes
(2)
C∗ (X) ' C∗ (X1 ∨ X2 ∨ . . . ).
Hence we get by (7.4) an isomorphism
(3)
Hn (X; M ) = Hn (X1 ∨ X2 ∨ . . . ; M )
M
M
Hn (Xi ; M ) ⊕
Hn+1 (Xi ∧ Xj ; Mee )
=
i<j
i≥1
In the second row we use the cross effect formula. Since we know Hn (Xi ) by (7.10)
we thus get a description for Hn (X; M ) as an abelian group. For this we point out
that for A, B, Mee ∈ Ab
(4)
Hn (M (A, i) ∧ M (B, j); Mee ) =





0
for
n<i+j
and
A ⊗ B ⊗ Mee
for
n = i + j,

Trp(A, B, Mee )



A ∗ B ∗ Mee
for
n = i + j + 1,
for
n = i + j + 2.
n > i + j + 2,
Here Trp is the triple torsion product of Mac Lane; see Notes on page 393 in [27].
Using (3) for M̂ in (7.7) we thus obtain by the exact sequence in (7.7) a possibility
to compute Hn (X, M ) for an arbitrary square group M in terms of the boundary
∂ : Hn (X, M add ) → Hn−1 (X, M̂ ). For example we obtain the following vanishing
theorem.
(7.13) Proposition. Let X be a connected CW-space with dim X = N and let M
be a square group. Then
Hn (X, M ) = 0
for
n ≥ 2N
and
Hn (X, M ) = Hn (X, M̂ )
for
N < n < 2N.
8. Stable square-homology and Steenrod squares
Given a square group M we obtain the square-homology Hn (X; M ) and the stable square-homology HnS (X; M ) which is the stabilization of Hn (X; M ) as defined
in (2.2). We have seen in (6.8) that the stable square-homology HnS (X; M ) is a
homology theory on space 1 . Hence by (2.8) there is a spectrum EM associated to
the square group M with
(8.1)
HnS (X; M ) = Hn (X; EM ).
This equation also holds if X is not a finite CW-space since both sides of the
equation satisfy the colimit axiom.
(8.2) Remark. The correspondence M 7→ EM yields a functor
E : Square → Ho(spectra)
where the right hand side is the homotopy category of spectra. To obtain this
functor we use Goodwillie’s equivalence [17] between linear homotopy functors and
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QUADRATIC HOMOLOGY
461
spectra. Hence the functor E is obtained by the functor which carries M to M]lin
where the linear homotopy functor M]lin determines functorially EM by
M]lin (X) = Ω∞ (EM ∧ X).
Compare (4.3) and (4.11).
For an abelian group A let K(A) be the Eilenberg-Mac Lane spectrum with
K(A)n = K(A, n). Let K(A)[i] be the shifted spectrum with K(A)[i]n =
K(A, n + i) where i ∈ Z. We clearly have
(8.3)
Hn (X; K(A)[i]) = H̃n−i (X; A)
where the right hand side is the reduced singular homology of X with coefficients
in A.
(8.4) Theorem. Let M be a quadratic Z-module. Then the stable square homology
with coefficients in M is given by an isomorphism, X ∈ space1 ,
HnS (X, M ) =
M
H̃n−i (X, Hi M∗e )
i≥0
where M∗e is the chain complex in (7.8). The isomorphism is natural in X. Hence
by (8.3) the spectrum EM associated to M is homotopy equivalent to the product of
shifted Eilenberg-Mac Lane spectra K(Hi M∗e )[i] with i ≥ 0.
Proof. We know by (4.11) that there exists a spectrum E with
M]lin (X) = Ω∞ (E ∧ X)
Hence E is determined by M]lin (S n ) where we can use a large dimension n. Now in
the stable range M] (S n ) → M]lin (S n ) is an equivalence. Here M] (S n ) is given by
an abelian simplicial group since M is a quadratic Z-module. An abelian simplicial
group, however, is equivalent to a product of Eilenberg-Mac Lane spaces. Hence
all k-invariants of M]lin (S n ) vanish and therefore E is a product of Eilenberg-Mac
Lane spectra.
(8.5) Theorem. Let M be a square group and let M add = cok P be defined by M .
Then the spectrum EM associated to M is homotopy equivalent to the cofiber of a
map between spectra
SqM : K(M add )[−1] → K(Hi M∗e )[i].
i≥1
Proof. For the proof we apply the Goodwillie calculus of functors which yields the
fibre sequence in (4.11) (2). Applying this fiber sequence to M] and M̂] in (7.7)
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462
HANS-JOACHIM BAUES
gives us the rows in the following commutative diagram of homotopy functors.
Ω M]add


y
F̂ −−−−→
k
F −−−−→
= K

µ
y
M̂]


y
−−−−→
M̂]lin


∗ y
M]


y
−−−−→
M]lin
M]add
Here the column in the middle is obtained by the fiber sequence in (7.7). Now
(4.11) (2) and (6.9) show that the fibres F and F̂ coincide so that the subdiagram
∗ is a pull back. Hence the fiber K of M̂]lin → M]lin coincides with
K = Ω M]add = K(M add )[−1]]
Here we denote by E] the homotopy functor given by a spectrum E with
E] (X) = Ω∞ (E ∧ X).
Now µ in the diagram yields by (8.4) the map SqM in the theorem since a fiber
sequence of spectra is also a cofiber sequence.
(8.6) Definition. The map SqM in (8.5) is a (multiple) cohomology operation which
we call the squaring operation associated to the square group M . For i ≥ 1 let
i+1
: K(M add )[−1] → K(Hi M∗e )[i]
SqM
i+1
depends on the choice
be the coordinate of SqM of degree i + 1. Clearly SqM
of the homotopy equivalence in (8.4). Below we show that the classical Steenrod
squares
Sq i+1 : K(Z/2)[−1] → K(Z/2)[i]
yield examples of such squaring operations. Moreover we compute the operation
2
SqM
for any M ∈ Square in the next section. Since in the bottom degree the
2
is actually independent
homotopy equivalence in (8.4) is canonical we see that SqM
2
is also natural in
of the choice of the homotopy equivalence in (8.4). Hence SqM
M.
It is a classical result that any short exact sequence A B C of abelian
groups induces a long exact sequence of homology groups:
β
→ Hn (X, A) → Hn (X, B) → Hn (X, C) → Hn−1 (X, A) →
where β is the Bockstein operator. In a similar way one has for each short exact
sequence L M N of square groups a long exact sequence of square-homology
groups:
(8.7)
β
→ Hn (X, L) → Hn (X, M ) → Hn (X, N ) −→ Hn−1 (X, L) → .
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QUADRATIC HOMOLOGY
463
This sequence is obtained in the same way as the special case in (7.7) (2). In
addition the stabilization of (8.7) yields the long exact sequence of stable squarehomology groups:
(8.8)
β
S
→ HnS (X, L) → HnS (X, M ) → HnS (X, N ) −→ Hn−1
(X, L) → .
Clearly the sequences (8.7) and (8.8) are natural with respect to maps between
short exact sequences in Square. They are also natural in X ∈ space 1 . As a
special case we obtain for M̂ M M add the commutative diagram:
β
S
(X, M add ) −−−−→
Hn+1
k
(8.9)
(SqM )∗
H̃n+1 (X, M add ) −−−−−→
HnS (X, M̂ )
L
k
H̃n−i (X, Hi M∗e )
i≥1
Here the bottom arrow is induced by the squaring operation SqM in (8.6). For this
recall that a map φ : E → E 0 between spectra induces a map φ∗ : Hn (X, E) →
Hn (X, E 0 ) between homology groups. See (2.8) and (8.3). In diagram (8.9) the
homomorphism β is natural in X and M . The isomorphism on the right hand side
of (8.9) is given by the choice in (8.4).
It is well known that the Steenrod square Sq 1 is obtained as a Bockstein operator;
that is,
(8.10)
β = (Sq 1 )∗ : Hn+1 (X, Z/2) → Hn (X, Z/2)
coincides with the Bockstein operator associated to Z/2 Z/4 → Z/2. The
next result shows that all Steenrod squares Sq i , i ≥ 1, are actually obtained by a
Bockstein operator of stable square-homology. For this we use the square group
0
Z4,2
nil = (Z/4 −→ Z/2 −→ Z/4)
H
(8.11)
with H{n} = {n(n − 1)/2}. Here {n} ∈ Z/k denotes the coset of n ∈ Z. For Znil
in (6.3) one has a canonical quotient map q : Znil → Z4,2
nil given by Z Z/4 and
Z Z/2. There is a short exact sequence in Square
(Z/2)Γ Z4,2
nil Z/2
(8.12)
given by the diagram
1
0
(Z/2)Γ = (Z/2 −−−−→ Z/2 −−−−→ Z/2)




k
y
y
0
Z4,2
nil = (Z/4


y
−−−−→ Z/2 −−−−→ Z/4)




y
y
Z/2 = (Z/2
−−−−→
0
−−−−→ Z/2)
One readily checks that for M = (Z/2)Γ one has Hi M∗e = Z/2 for i ≥ 0. Hence
one gets by (8.4) a Bockstein operator β as in the following theorem.
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464
HANS-JOACHIM BAUES
(8.13) Theorem. Let β be the Bockstein operator of stable square homology associated to the short exact sequence in (8.12). Then the following diagram commutes:
β
S
Hn+1
(X, Z/2) −−−−→ HnS (X, (Z/2)Γ )
k
k
Sq
H̃n+1 (X, Z/2) −−−−→ ⊕ H̃n−i (X, Z/2)
i≥0
Here Sq has the coordinates (χSq i+1 )∗ induced by the Steenrod squares Sq i+1 for
i ≥ 0 where χ is the anti-automorphism of the Steenrod algebra; see 27.24 [22].
Moreover the isomorphism on the right hand side of the diagram is obtained by an
appropriate choice of the homotopy equivalence in (8.4).
For the proof of this theorem we use the mod-2 restricted lower central series
Γ̄n (G) which defines the group functor
nil2 : Gr → Gr
(8.14)
with nil2 (G) = G/Γ̄3 (G)
This is the restricted version of nil2 in (5.11). For a free group G we have the
following facts (1), (2) and (3).
G/Γ̄2 (G) = Gab ⊗ Z/2 = V
(1)
is a Z/2-vector space and
(2)
Γ̄2 (G)/Γ̄3 (G) = L̄2 (V ).
Here L̄2 (V ) ⊂ V ⊗ V is generated by [v, w] = v ⊗ w + w ⊗ v and v ⊗ v for v, w ∈ V .
L̄2 (V ) is the degree 2 part of the free mod-2 restricted Lie algebra L̄(V ). Using
7.6 [15] one obtains (1) and (2) above. Hence one obtains the natural short exact
sequence in the top row of the following commutative diagram. The bottom row is
obtained by the exact sequence in (8.12) and the quadratic tensor product.
0 −−−−→ L̄2 (Gab ⊗ Z/2) −−−−→ nil2 (G) −−−−→ Gab ⊗ Z/2 −−−−→ 0
k
(3)
0 −−−−→
G ⊗ (Z/2)Γ
k
k
−−−−→ G ⊗ Z4,2
nil −−−−→ G ⊗ Z/2 −−−−→ 0
The vertical isomorphisms which are natural in G are induced by G⊗Znil = nil2 (G)
in (6.4); see 8.1 [10].
Proof of (8.13). If G = α(X) is the free simplicial group given by X then the exact
sequence of simplicial groups in (8.14) (3) induces the Bockstein operator in (8.13).
The connecting homomorphism
d1 : πn (Gab ⊗ Z/2) → πn−1 L̄2 (Gab ⊗ Z/2)
is computed in 8.10 [15] in terms of the Steenrod operations Sq i which act from
the right on homology since for a finite type space X we have H∗ (X, Z/2) =
Hom(H ∗ (X, Z/2), Z/2). The anti-isomorphism χ of the Steenrod algebra has the
property that for y ∈ H∗ (X, Z/2) one has
(χSq i )∗ (y) = y Sq i
Therefore the stabilization of the differential d1 in 8.10 [15] yields the result.
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QUADRATIC HOMOLOGY
465
As an application of (8.13) and (8.5) we obtain the following results which determine explicitly the spectra EM associated to M = Znil and M = Z4,2
nil respectively.
For this we consider the following commutative diagram in Square:
0 −−−−→
(8.15)
ZΛ


y
−−−−→ Znil −−−−→


y
Z


y
−−−−→ 0
0 −−−−→ (Z/2)Λ −−−−→ Z4,2
nil −−−−→ Z/4 −−−−→ 0






y
y
y
0 −−−−→ (Z/2)Γ −−−−→ Z4,2
nil −−−−→ Z/2 −−−−→ 0
Here the rows are exact and the vertical arrows are quotient maps. The top row and
the row in the middle are the exact sequences M̂ M M add for M = Znil and
Λ
Λ
M = Z4,2
nil respectively. We have Z = (0 → Z → 0) and (Z/2) = (0 → Z/2 → 0).
Naturality of the Bockstein operator yields by (8.9) a relation of SqM for M = Znil
Λ
Λ
Γ
and M = Z4,2
nil with β in (8.13). Since the maps Z → (Z/2) → (Z/2) induce
injections
H∗ (ZΛ )e∗ H∗ (Z/2Λ )e∗ H∗ (Z/2Γ )e∗
(see (7.8)) we get the following results.
(8.16) Theorem. For the square group M = Znil we have the spectrum E = EM
with
HnS (X, nil2 ) = HnS (X, Znil ) = Hn (X, E).
This spectrum E is homotopy equivalent to the cofiber of the map
SqM : K(Z)[−1] → × K(Z/2)[i].
i odd
∗
Here the coordinate of degree i + 1 is q (χ Sq i+1 ) where q : Z Z/2 is the quotient
map and χ is the anti-automorphism of the Steenrod algebra.
(8.17) Theorem. For the square group M = Z4,2
nil we have the spectrum E = EM
with
S
Hm
(X, nil2 ) = HnS (X, Z4,2
nil ) = Hn (X, E).
This spectrum E is homotopy equivalent to the cofiber of the map
SqM : K(Z/4)[−1] → × K(Z/2)[i].
i≥1
∗
Here the coordinate of degree i + 1 is q (χ Sq i+1 ) where q : Z/4 → Z/2 is the
quotient map and χ is the anti-automorphism of the Steenrod algebra.
2
9. The squaring operation SqM
2
The squaring operation SqM
associated to a square group M is an element in
the following group where n is large; see (8.6).
[K(M add )[−1], K(H1 M∗e )[1])
= [K(M add , n − 1), K(H1 M∗e , n + 1)]
(9.1)
= Hom(M add ⊗ Z/2, H1 M∗e ).
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466
HANS-JOACHIM BAUES
2
Here we have M add = cok (P ) and H1 M∗e = ker (P )/im (2 − HP ) so that SqM
is a
homomophism
2
SqM
: cok (P ) ⊗ Z/2 → ker (P )/im (2 − HP ).
We now consider the following diagram where ( | )H is the cross effect of H in
(6.3) and where ∆ is the diagonal with ∆(x) = (x, x) for x ∈ Me .
(9.2)
Me


∆y
q0 ⊗1
−−−−→
Me × Me −−−−−−→
(
|
)H
Sq2
M
cok (P ) ⊗ Z/2 −−−−
→ ker (P )/im (2 − HP )
x
q

⊃
Mee
ker (P )
Here q is the quotient map and q 0 : Me → cok (P ) is the quotient map.
2
(9.3) Theorem. For a square group M there is a unique homomoprhism SqM
such
that diagram (9.2) commutes and this homomorphism coincides with the squaring
2
by use of (9.1).
operation SqM
Proof of (9.3). There exist in fact only two homomorphisms from Me to ker(P )/
im(2 − HP ) which are defined for all square groups M and which are natural in
2
2
M . We have seen that SqM
in (8.6) is natural in M . Moreover SqM
in (8.6) is
2
0
nontrivial by (8.16). But the nontrivial natural map is SqM (q ⊗ 1) in (9.2) so that
2
2
SqM
has to coincide with SqM
in (9.2).
2
2
For example for M = Znil the homomorphism SqM
is the isomorphism SqM
:
Z⊗Z/2 ∼
= Z/2. We now give an interpretation of diagram (9.2) in terms of algebraic
models of (n − 1)-connected (n + 1)-types, n ≥ 2, obtained in [6]. This leads to an
additional proof of (9.3).
(9.4) Definition. A reduced quadratic module (ω, δ) is a diagram
ω
δ
M ab ⊗ M ab −→ L −→ M
of homomorphisms between groups such that the following properties hold. The
groups M, L have nilpotency degree 2 and the quotient map M → M ab is denoted
by x 7→ {x}. Then for x, y ∈ L, a, b ∈ M we have
δω({a} ⊗ {b}) = −a − b + a + b,
ω({δx} ⊗ {δy}) = −x − y + x + y,
ω({δx} ⊗ {a} + {a} ⊗ {δx}) = 0.
We say that (ω, δ) is a stable quadratic module if in addition
ω({a} ⊗ {b} + {a} ⊗ {b}) = 0
A morphism (ω, δ) → (ω 0 , δ 0 ) is a commutative diagram in Gr
l
L −−−−→


δy
L0

0
yδ
m
M −−−−→ M 0
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QUADRATIC HOMOLOGY
467
such that l ω = ω 0 (mab ⊗ mab ). This is a weak equivalence if (l, m) induces isomorphisms
ker (δ) ∼
= ker (δ 0 ), cok (δ) ∼
= cok(δ 0 ).
Let rquad (resp. squad) be the categories of reduced (resp. stable) quadratic
modules and let Ho rquad, Ho squad be the localization with respect to weak
equivalences. There are equivalences of categories for which the following diagram
commutes, n ≥ 3; see [6] and [8].
λ
types12 −−−2−→ Ho rquad
∼
Ω
(9.5)
n−2
∪
∪i
λn
types1n −−−−→ Ho squad
∼
Here types1k is the homotopy catgeory of CW-spaces X for which πi X = 0 for
i < k and i > k + 1. Moreover Ωn−2 is the iterated loop space functor and i
is the inclusion functor. For X ∈ types1n and (ω, δ) = λn (X) we have natural
isomorphisms
(1)
πn X = πn = cok(δ), πn+1 (X) = πn+1 = ker(δ).
Moreover the k-invariant of X is an element
(2)
k(X) ∈ H n+2 (K(πn , n), πn+1 ) = Hom(Γ1n (πn ), πn+1 ).
Here Γ1n is the functor Ab → Ab which is Whitehead quadratic functor Γ for n = 2
and ⊗ Z/2 for n ≥ 3. We can obtain k(X) from (ω, δ) = λn (X) as follows. Given
(ω, δ) ∈ rquad there is a unique homomorphism k for which the following diagram
commutes:
q∗
Γ(M ab )
/
Γ(cok δ)
k
/
H
ω
M ab ⊗ M ab
ker δ
∩
L
δ
/
/
M
Here H is the cross effect map in (3.1) for the functor Γ and q∗ is induced by the
quotient map M → M ab → cok δ which factors through M ab . If (ω, δ) is stable
then k admits a factorization
(3)
k0
σ
k : Γ(cok δ) −→ cok(δ) ⊗ Z/2 −→ ker δ.
Now the k-invariant k(X) with (ω, δ) = λn X is k for n = 2 and is k 0 for n ≥ 3.
Compare [6, 8].
Recall that Square is the category of square groups in (6.3). We now define
canonical algebraic functors
τ
L
Square −−−−→ rquad −−−−→ Ho rquad
∪
(9.6)
τ
∪
lin
Square −−−−→ squad −−−−→ Ho squad
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468
HANS-JOACHIM BAUES
together with a natural transformation a : τ → τ lin . Here L is the localization
functor. The functor τ is given by
ω
δ
τ (M ) = (Meab ⊗ Meab −→ Mee −→ Me )
where ω({x} ⊗ {y}) = (x|y)H and δ = P . Moreover let
ω0
δ0
τ lin (M ) = (Meab ⊗ Meab −→ Mee /im (2 − HP ) −→ Me )
where ω 0 , resp. δ 0 , are induced by ω, resp. δ, in τ (M ) above. We clearly have the
natural map a : τ (M ) → τ lin (M ) which is the identity on Me and the quotient
map on Mee . One readily checks that the functors τ and τ lin are well defined; for
this we only prove
(9.7) Lemma. τ lin (M ) is a stable quadratic module.
Proof. We have to show that for a, b ∈ Me we have (a|b)H + (b|a)H ∈ im(2 − HP ).
But we know by 3.5 (4) in [10] that ∆ : Me → Mee with
∆(a) = (HP − 2)Ha + (a|a)H
is a homomorphism. Hence we get
(a|b)H + (b|a)H = (a + b|a + b)H − (a|a)H − (b|b)H
=
∆(a + b) + (2 − HP )H(a + b)
− (∆(a) + (2 − HP )H(a) + ∆(b) + (2 − HP )H(b))
=
(2 − HP )(a|b)H .
This term needs not to be trivial in Mee as the example Znil shows. Hence τ (M )
in general is not stable.
We now obtain functors
(9.8)
S, S lin : Square → types12
which carry the square group M to S(M ) = M] (S 2 ) and S lin (M ) = 3-type of
M]lin (S 2 ) respectively. Here (7.10) shows that M] (S 2 ) ∈ types12 . We have the
obvious map
b : S(M ) = M] (S 2 ) → M]lin (S 2 ) → S lin (M )
which is natural in M . We also observe that S lin (M ) is an infinite loop space since
M]lin (S 2 ) is an infinite loop space.
(9.9) Theorem. For the functors in (9.8), (9.6) and (9.5) there exist natural isomorphisms such that the following diagram commutes.
∼
L τ (M )
λ2 S(M )
=


a

b∗ y
y ∗
λ2 S lin (M )
∼
=
L τ lin (M )
2
in (9.2) is the k-invariant of τ lin (M ).
Proof of (9.3). One readily checks that SqM
lin
The k-invariant of τ (M ) coincides with the first non-trivial k-invariant of
M]lin (S 2 ) by (9.9). This yields the proposition in (9.3).
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QUADRATIC HOMOLOGY
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Proof of (9.9). We have by (7.10) that
π2 (S(M )) = π2 M ] (S 2 ) = Z ⊗ cok(P ) = cok(P ),
π3 (S(M )) = π3 M ] (S 2 ) = Z ⊗ Z1 M∗ = ker(P ).
∼ L τ (M ) exists on the level of homotopy groups.
Hence the isomorphism λ2 S(M ) =
An explicit isomorphism is given by the result of Conduché [14]. For this let G
be the free simplicial group generated by one element in degree 1. Then we have
G ' α(S 2 ) in (5.1) so that M ] (S 2 ) ' B|G ⊗ M |. Here G ⊗ M is a simplicial group
with only two nontrivial homotopy groups π1 and π2 . Hence the homotopy type
of G ⊗ M is described by its reduced 2-module in the sense of Conduché (see (2.9)
[14]). But this reduced 2-module of G ⊗ M coincides with τ (M ). This yields the
isomorphism λ2 S(M ) ∼
= L τ (M ). Moreover we know that a∗ coincides with b∗ on
the level of homotopy groups since
π3 (S(M ))


b∗ y
=
ker(P )

a
y
π3 (S lin (M ))
=
ker(P )/in(2 − HP )
commutes. This shows that the k-invariant of S lin (M ) is actually the k-invariant
of τ lin (M ). Moreover the natural isomorphism λ2 S(M ) ∼
= L τ (M ) induces the
natural isomorphism λ2 S lin (M ) ∼
= L τ lin (M ).
Appendix A: A criterion for 2-excisive functors
We describe a criterion for 2-excisive functors which shows that only very special
strongly co-Cartesian diagrams are needed to determine a 2-excisive functors. We
use this result for the proof of theorem (6.6). It would be interesting to obtain a
similar result for n-excisive functors.
Let Y be a 2-cube consisting of spaces Y1 , Y2 , Y3 , Y4 and let A ∈ space. Then
we define A ∨ Y by
A ∨ Y1 −−−−→ A ∨ Y3




y
y
A ∨ Y2 −−−−→ A ∨ Y4
This yields the 3-cube (0, 1) : A ∨ Y → Y . Clearly if Y is co-Cartesian then
(0, 1) : A ∪ Y → Y is a strongly co-Cartesian 3-cube. Recall the definition of
Cube (X, A) in the proof of (4.12).
(A.1) Theorem. Let D : space r → space 1 , r ≥ 0, be a homotopy functor.
Assume that for pairs (X, A) ∈ pair r and all homotopy push outs Y in space r
the 3-cubes D(Cube (X, A)) and D((0, 1) : A ∨ Y → Y ) are Cartesian. Then the
homotopy functor D is 2-excisive.
The theorem shows that essentially the 3-cubes Cube (X, A) suffice to determine quadratic homotopy functors. This again shows that the quadratic excision
sequence essentially covers all excision properties of a 2-excisive functor.
Proof of (6.6). Let F be a quadratic coefficient functor so that F = ⊗M by (6.4).
Since we can use (5.4) it is enough to prove that F] is 2-excisive. For this we
can use theorem (A.1) above. Using (1) in the proof of (6.9) above we see that
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470
HANS-JOACHIM BAUES
F] ((0, 1) : A ∨ Y → Y ) is Cartesian if Y is co-Cartesian. Moreover using the pull
back diagram (4) in the proof of (6.9) we see that F] (Cube(X 0 , A0 )) is Cartesian
for (X 0 , A0 ) ∈ pair 1 . For this we use (4.8). Hence the assumptions of (A.1) are
satisfied and therefore F] is 2-excisive.
Proof of (A.1). Let Y be a cofibrant co-Cartesian 2-cube and let i : Y ⊂ A ∨ Y
be the inclusion then D(i) is Cartesian since D(0, 1) in (6.8) is Cartesian and
(0, 1)i = 1. For this we use the lemmas in section 1 of [12]. Next let (Y1 , A) be
a pair in pair r . Then we obtain the 3-cube q : Y → Y /A given by the quotient
maps qi : Yi → Yi /A. Clearly q is strongly co-Cartesian. For this we put the cubes
Cube (Yi , A) and the cube (0, 1) : A ∨ Y → Y together to form a large cube with
boundary q : Y → Y /A. We apply D to this large cube and then we take homotopy
fibers which form the following commutative diagram where Qi is the homotopy
fiber of (qi )∗ : D(Yi ) → D(Yi /A).
(1)
Q1
JJ
JJ
JJ
JJ
JJ
/
tt
tt
tt
t
t
tt
e
9
D(A ∨ Y1 )2
D(A ∨ Y2 )2
/
D(A ∨ Y3 )2
/
D(A ∨ Y4 )2
JJ
JJ
JJ
JJ
JJ
t
tt
tt
t
tt
tt
y
Q3
Q2
%
/
Q4
Here the inside square is Cartesian since D(0, 1) is Cartesian by the assumption in
(A.1). Moreover, since D(Cube (Yi , A)) is Cartesian by the assumption in (A.1) we
see that the four boundary subdiagrams are Cartesian by (4.8). For this we use the
following diagram; compare the proof of (4.9).
K(A)


y
(2)
−−−−→
K0


y
−−−−→
K 00


y
D(A ∨ A)2 −−−−→ D(A ∨ Y1 )2 −−−−→ D(A ∨ Y2 )2






∗
y
y
y
D(A)
−−−−→
Q1
−−−−→
Q2
Here the columns are fiber sequences. We know that K(A) → K 0 and K(A) → K 00
are homotopy equivalences. Hence also K 0 → K 00 is a homotopy equivalence and
therefore the subdiagram * is Cartesian. Now all subdiagrams of (1) being Cartesian
we see that also the boundary diagram of (1) is Cartesian. This completes by (4.8)
the proof that D(q : Y → Y /A) is Cartesian. Now let Y ∪ CA be the 2-cube given
by Yi ∪A CA. Since the quotient map Yi ∪CA → Yi /A is a homotopy equivalence we
see that also the inclusion j : Y ⊂ Y ∪ CA is a 3-cube for which D(j) is Cartesian.
Now let X and Y be 2-cubes and let f : Y → X be a strongly co-Cartesian 3cube. We may assume up to equivalence that f is cofibrant, that is all maps in
the 3-cube f are cofibrations and all subsquares are actually push outs. Using
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QUADRATIC HOMOLOGY
471
CW-decomposition we can filter f by an infinite sequence
f : Y = X 0 ⊂ X 1 ⊂ X 2 ⊂ . . . ⊂ colim(X i ) ' X.
Here all X i ⊂ X i+1 are cofibrant co-Cartesian 3-cubes for which there exist Ai ∈
space r such that X 1 = Y ∨ A1 and Xi+1 = Xi ∪ CAi , i ≥ 1. We have seen above
that D(X i → X i+1 ) is Cartesian for all i ≥ 0. This implies by I.4b.5 in [3] that also
D(Y → X) is Cartesian since we assume that D preserves filtered colimits.
Appendix B: The homology spectral sequence
Let D be a reduced homotopy functor. We describe a spectral sequence which
converges to the homology with coefficients in D. If D is linear this is the AtiyahHirzebruch spectral sequence.
(B.1) Definition. Let D : space 1 → space 1 be a reduced homotopy functor and
let X be a connected CW-complex. Let U = C(X) be the reduced cone on X
which is filtered by X = U 0 ⊂ U 1 ⊂ . . . ⊂ U with U n = X ∪ C(X n−1 ), n ≥ 1. For
(Y, B) ∈ pair 1 let
Hn (Y, B) = Hn (Y, B; D)
(1)
be the D-homology in (4.2). Then the filtered space U gives rise to the following
spectral sequence. First we can form the long exact sequence
j
(2)
∂
. . . → H2 (U n ) −→ H2 (U n , U n−1 ) −→ H1 (U n−1 )
j
i
−→ H1 (U n ) −→ H1 (U n , U n−1 ) → 0
where the last object is a set with basepoint 0 and all the other objects are groups.
We can form the rth derived homotopy sequences of (2) for r ≥ 0, n ≤ 0,
(r)
(3)
(r)
(r)
. . . → H2 (n − 2r − 1) → F1 (n − r) → H1 (n − r)
(r)
(r)
→ H1 (n − r − 1) → F0 (n) → 0.
Here we set for q ∈ Z, n = −m,
Hq(r) (n) = image(Hq (U m−r ) → Hq (U m )),
Fq(r) (n) = kernel(Hq+1 (U m+1 , U m ) → Hq (U m )/Hq(r) (n))/
action of kernel(Hq+1 (U m+1 ) → Hq+1 (U m+r+1 )).
For r = 0 the sequence (3) coincides with (2). One can check that (3) is well defined
and has the same properties as (2). Let
(4)
(r−1)
Ers,t = Ft−s (s)
for
t ∈ Z, s ≤ 0, r ≥ 1,
and let the differential dr : Ers,t → Ers+r,t+r−1 of degree (r, r−1) be the composition
(r−1)
(r−1)
(r−1)
F(t−s) (s) → Ht−s (s) → Ft−s−1 (s + r)
where we use the operators from (3). Clearly we have for q ≥ 0, s ≥ 0
(5)
E1−s,q−s = Fq(0) (−s) = Hq+1 (U s+1 , U s )
and d1 is the composition
(6)
d1 = j∂ : Hq+1 (U s+1 , U s ) → Hq (U s ) → Hq (U s , U s−1 ).
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HANS-JOACHIM BAUES
Assume that the pairs (U m , U m−1 ) have the property that there exist 0 ≤ N0 ≤
N1 ≤ . . . with lim {Nm} = ∞ such that
(7)
Hi (U m+1 , U m ) = 0
for
i < Nm .
Then we can find for each q ≥ 0 a bound r = r(q) < ∞ such that
(8)
s,q+1
s,q+s
= . . . = E∞
.
Ers,q+s = Er+1
We obtain a filtration of Hq (X) for q ≥ 1, s ≥ 0,
0 = K0,q ⊂ K1,q ⊂ . . . ⊂ Ks,q ⊂ . . . ⊂ Hq (X),
Ks,q = kernel (Hq (X) → Hq (X/X s )).
S
Since D is reduced we see that s≥0 Ks,q = Hq (X) and
(9)
−s,q−s
E∞
= Ks,q /Ks+1,q .
(10)
A similar spectral sequence is available for reduced homotopy functors which map
to spectra; compare (4.1).
(B.2) Remark. The spectral sequence Ers,t above coincides with the spectral sequence in (III.10.4) [3] by considering the relative homotopy groups of the filtered
space {D(U n )}. The conventions for indexing Ers,t arises from the comparison with
the Bousfield-Kan spectral sequence [12]; compare the discussion in (III.10.2) [3].
We can alter the indexing by defining
r
= Er−s,t ,
Es,t
s ≥ 0, t ∈ Z.
has degree (−r, r − 1) as in the spectral sequence
Then the differential for
discussed by G.W. Whithead, chapter XIII, page 614 [29]. Now assume that D in
(B.1) is a linear homotopy functor. Then Hn (Y, B; D) = Hn (Y, B) is a homology
theory on pair1 in the sense of (2.2) and we get for s ≥ 0
r
Es,t
1
= E1−s,q−s = Hq+1 (U s+1 , U s )
Es,q−s
= Hq+1 (X ∪ CX s , X ∪ CX s−1 )
= Hq+1 (ΣX s , ΣX s−1 ) = Hq (X s , X s−1 ).
Moreover by the exact sequence
Hq (X s ) → Hq (X) → Hq (X/X s ).
We see that K−s,q in (B.1) (9) coincides with Is,q−s where Is,q−s is the image of
Hq (X s ) → Hq (X) as on page 613 [29]. This shows that the spectral sequence (B.1)
yields as a special case the Atiyah-Hirzebruch spectral sequence in XIII.3.3 [29].
Appendix C: A spectral sequence for square homology
In this section we apply the spectral sequence of Appendix B to square homology.
Let X be a CW-complex with trivial 0-skeleton X 0 = ∗ and let M be a square group.
We consider the filtration, s ≥ 0,
0 = K0,q ⊂ K1,q ⊂ . . . ⊂ Ks,q ⊂ . . . ⊂ Hq (X; M )
of square homology given by
(C.1)
Ks,q = kernel (Hq (X; M ) → Hq (X/X s ))
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QUADRATIC HOMOLOGY
473
where X → X/X s is the quotient map. Then the spectral sequence (Er−s,q−s , dr )
defined in the Appendix B converges and satisfies
−s,q−s
E∞
= Ks,q /Ks−1,q .
(C.2)
We now determine the E1 -term of this spectral sequence. Let
U s = X ∪ CX s−1 ' X/X s−1
and let As = X s /X s−1 . Then we have
U s+1 = X ∪ CX s+1 = U s ∪ CAs
(C.3)
where the attaching map As → U s is the composite f : As ⊂ X/X s−1 ' U s given
by the inclusion. Clearly As = M (Cs , s) is the Moore space of the free abelian group
Cs + Hs (X s , X s−1 ; Z) which is part of the cellular chain comlex C∗ X = (C∗ , d).
We can compute the E1 -term
E1−s,q−s = Hq+1 (U s+1 , U s ; M )
(C.4)
by applying diagram (3.4) to (C.3). This yields the following commutative diagram
in which the row and the column are exact.
Hq+1 (As ∧ As ; Mee )
(C.5)
(P,−(1,f )∗ )
Hq (As ; M ) ⊕ Hq+1 (As ∧ Xs ; Mee )
UUU
UUU(f ,P (f,1) )
∗
U∗UU
UUU
UUU
UU
*
Hq+1 (U s+1 , U s ; M )
∂
/
Hq (U s , M )
Hq (As ∧ As ; Mee )
(P,−(∗1,f )∗ )
Hq−1 (As ; M ) ⊕ Hq (As ∧ Xs ; Mee )
Using (7.10) this yields the following result on Hq+1 (U s+1 , U s ; M ). Let Bs =
image(d : Cs+1 → Cs ) be the group of boundaries in the cellular chain complex
C∗ X and let q : Cs → Cs /Bs be the quotient map.
(C.6) Proposition. The E1 -term is given by the groups
scribed by the following isomorphisms and exact sequences

0 for


s+1
s
Cs ⊗ Hq−s M∗e for
, U ; M) =
Hq+1 (U

C ⊗ H
s
q−s+1 (X; Mee ) for
(C.4) which can be derespectively.
q < s,
s ≤ q < 2s − 1,
q > 2s.
For q = 2s − 1 and q = 2s one has the exact sequence
j
0 → Cs ⊗ Hs+1 (X; Mee ) −→ H2s+1 (U s+1 , U s ; M ) → Cs ⊗ Cs ⊗ Mee
(P,−q∗ )
−→ C̄s ⊗ Zs−1 M∗ ⊕ Cs ⊗ Cs /Bs ⊗ Mee → H2s (U s+1 , U s ; M ) → 0.
Here C̄s = Cs for s > 1 and C̄1 = π1 (X 1 ) is the fundamental group of the 1-skeleton
X 1 for s = 1 and Z0 M∗ = M .
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474
HANS-JOACHIM BAUES
A cross effect argument shows that the inclusion j in the exact sequence of (C.6)
is split injective. Moreover the differential d1 of the spectral sequence is of the
form d ⊗ 1 (with d given by C∗ X) for the groups in the first part of (C.6) with
q 6= 2s − 1, 2s. It is interesting to compare (C.6) with the results in section 7.
This shows that the spectral sequence for q > s depends only on M̂ and hence is
determined in this range by C∗ X.
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Max-Planck-Institut für Mathematik, Gottfried-Claren-Strasse 26, D-53225 Bonn,
Germany
E-mail address: [email protected]
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